Lensball

Lensball
A cap of the source $A$ is wrapped onto the lens $B$, drawn as a movable scope. Here the 60° cap fills all of $B$ by an equal-area wrap; the seam is at the back. Dashed lines: where the cap boundary goes. Rings: lens centre $c$ and its image $b_0$.
Kind	Stateful local representation with an explicit transition rule
Source space	$A$ (here $S^2$, radius 2)
Lens space	$B$ (here $S^2$, radius 1); usually the sphere $S^2$, optionally the solid ball $D^3$ in volumetric variants
Lens state	frame $Q\in SO(3)$, centre $c=RQ{e}_3$
View map	${\exp }_{b_0}\circ (\eta r/R)P\circ {\log }_c$ (declared for the spherical case)
Transition rule	$Q\leftarrow Q\exp (\kappa \mathrm{\Omega })$; equal-angle or equal-arc
Tiers	lensball · spherical lensball · discrete lensball
Key limits	invariance of domain · cut locus · holonomy (area/$R^2$)
Parent category	Interactive visualization lens (Tominski et al., 2017)
Term and formalisation	this wiki, 2026 (see positioning)
heidict	lensball (mirrored below)
Prototypes	~/ht/lensball-viz (numpy/matplotlib · Canvas/JS)

A lensball is a stateful local representation of a source space $A$ on a lens space $B$, usually a sphere, together with an explicit transition rule that is part of the model: interaction with the lens state induces navigation through the represented neighborhoods, slices, or projections of $A$. It aims to preserve selected local structure and temporal continuity rather than requiring one globally faithful low-dimensional embedding. The visible representation is therefore local and state-dependent: at any instant the lensball displays a selected neighborhood, slice, or projection of $A$; motion of the lensball changes that selection.

The central idea is not to require one static low-dimensional picture to preserve an entire high-dimensional space. Instead, a lensball aims for:

1. local geometric fidelity — the currently viewed region should preserve the geometry judged important;
2. exact identity where possible — a displayed datum can retain an exact link to its original representation in $A$;
3. temporal stability — nearby lensball states should produce nearby visual states rather than arbitrary rearrangements; and
4. navigability — rotations or other motions of $B$ induce controlled motion through the family of views of $A$.

In this sense, a lensball is closer to a movable chart, lens, or scope than to an ordinary global dimensionality-reduction map. These aims are shared with the interactive-lens literature; what is specific to lensball is stated under Positioning.

Figure M1. (Realised: infobox animation and Figure 2 above.) The basic lensball picture: a large source space $A$, a currently selected region $B^{\prime }\subset A$, and a smaller spherical view $B$. Rotating $B$ moves $B^{\prime }$ through $A$, while points displayed on $B$ retain links to their original points in $A$.

Illustrations

Both animations are the offline numpy/matplotlib prototype (spitball, 2026-09-14). Panels are, left to right: the source $A$ with the current chart outlined and the lens frame drawn (red up, blue right); the half-display view on $B$ through $\exp \circ \text{scale}\circ \log$; and the full-display view that uses all of $B$ minus one seam point.

Figure 1. $S^1\to S^1$, $R=2$, $r=1$. Segments: navigation along $A$; zoom $\eta$ stretching a shorter arc over $B$; navigation while $\eta$ ramps from 1 to 2 (equal-angle rule, no permutation between frames). The full-arc panel maps the same arc onto the whole circle with the seam at $-b_0$.

Figure 2. $S^2\to S^2$, $R=2$, $r=1$. Segments: navigation (the 60° cap slides over $A$); zoom with cap $60^{\circ }/\eta$ at constant display extent (dots spread: sparsity is information); parallel transport around an octant loop, after which the centre and cap are identical but the frame is rotated by exactly 90° = enclosed area / $R^2$ (Gauss–Bonnet). The full-display panels use the equal-area profile $\cos \psi =1-2(1-\cos \vartheta )/(1-\cos \alpha )$, shown from the front and from the back with the seam centred; the same points cover all of $B$ at one quarter of the packed density, and the sample lattice shows spokes near the seam because the map's anisotropy is unbounded there although its area distortion is exactly zero.

In software

Open the interactive embedding lensball — a frozen wiki-article embedding with drag navigation, exact record lookup, a local distortion readout, and a clickable global preview minimap.

As a tool, a lensball is an interactive viewer for a large or high-dimensional dataset that shows one local view at a time on a small movable display, usually a rendered sphere, and recomputes that view as the user moves the lens. In practice it is almost always the discrete lensball: displayed markers keep exact references to their source records, and only the geometry of the view is approximate.

Oslo world model

Video: Oslo through a lensball (44 seconds; 28 September 2026; not published here). The local Oslo world model supplies actual terrain triangles, building walls and roofs, and the Sinsen Panorama mesh. The tour moves from Sinsen toward Torshov, inspects the raised geometry, expands the source neighborhood over central Oslo and the fjord, and turns the display to expose the reserved back seam. The recording uses neighborhood labels and omits street addresses, home markers and coordinate readouts. Short on-screen titles introduce each step; an optional English caption track adds brief explanations. The video is silent.

The reference surface uses Figure 2’s equal-area profile, with $\alpha =\rho /R$. Each source vertex’s ground position is mapped onto the lens sphere; its elevation extends radially, with central scale $s=1/(R\sin (\alpha /2))$. The existing perspective model’s curvature/refraction drop is undone before radial elevation. Source orientation follows $Q\leftarrow Q\exp (\kappa \mathrm{\Omega })$, with the declared zoom-dependent sensitivity $\kappa =\alpha /2$. Display inspection changes only the camera; traversal changes the source chart. This is an elevated three-dimensional chart: equal area applies to the reference surface, not to building volumes or terrain relief.

The implementation retains source coordinates and mesh-triangle identities, while rasterization approximates the nonlinear map with transformed triangles. The region $\sin (\vartheta /2)/\sin (\alpha /2)\ge 0.995$ is omitted at the back seam. Distortion grows toward this region; the video does not imply global metric preservation. Game interiors and separate tree and sculpture meshes are not included in this demonstration.

The recording is not published on this wiki. Terrain: © Kartverket (CC BY 4.0); map data: © OpenStreetMap contributors (ODbL). Source and capture script: ~/world-model/view/web/lensball/ and view/tools/record_lensball.py.

Parameters

A concrete implementation is specified by choosing each component of the tuple in §1 of the mathematics section. The choices that matter most during development are:

Parameter	Role	Typical choice
Source metric $d_A$	what "near" means in the data	Euclidean, cosine, geodesic on a kNN graph
Lens display $B$	rendered surface	$S^2$; occasionally a disk or a solid ball
Lens state $\theta$	what the user moves	centre plus orientation ($Q\in SO(3)$), or a subspace $V$
Visible region $U_{\theta }$	which records are drawn	radius $\rho$, fixed point count $k$, or slice thickness $\delta$
Zoom $\eta$	angular magnification	1 to 10; bound the domain by $\vartheta <\pi /\eta$
Sensitivity $\kappa$	how far one gesture moves the lens	equal-angle or equal-arc-length (§4)
Layout objective	how visible points are placed	local stress plus temporal penalty, weights $\lambda ,\mu ,\nu$ (§13)
Orientation policy	what "up" means after motion	parallel transport, global reference, or user-controlled (§9)
Transition rule $T$	how control input changes $\theta$	the one part that must be explicit; it is the product, not a detail

Properties to test for

• Exact round trip. Selecting a marker returns the original record. This is a lookup, not a geometric inverse, and should never fail.
• Local stress. Distances among visible points match the source metric up to a scale, within a tolerance the task sets. Measure only on the visible set.
• Temporal stability. A small motion of the lens moves each shared marker by a small amount. Two views can each be good and still form a bad tool if the second permutes the first.
• Warm start. Each frame's layout is initialised from the previous frame. Cold recomputation is the usual cause of failed temporal stability.
• Determinism and path dependence. Decide whether returning to the same centre restores the same view. With parallel transport it does not, by design; state this in the interface.
• Controllability. If the state space is larger than the control space, such as a Grassmannian of slices driven by ball rotation, check that the transition rule actually reaches the states users need (§§10–11).

Numerical notes

• Compute the angle in the logarithm map with $\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{2}$ of the tangential and radial components rather than $\arccos (c\cdot p/R^2)$, which loses precision near the centre and the antipode.
• Use the series for $\vartheta /\sin \vartheta$ when $\vartheta$ is small; the closed form is 0/0 at the centre.
• Keep the visible region strictly inside the cut locus: never draw points near $-c$, where the map is undefined and neighbours are torn apart.
• Store orientation as a unit quaternion or re-orthonormalise $Q$ after each update; accumulated products drift out of $SO(3)$.
• Choose the lens radius from the error budget: distortion grows like $(L/R{)}^2$ for a curvature scale $R$, so halving the radius quarters the error and roughly quadruples the number of views needed to cover a surface (§8).
• For thick slices, tune $\delta$ against sample density; an exact slice of a finite dataset is empty.
• Do not expect a global fix. The Johnson–Lindenstrauss lower bound (§12) says a 2D or 3D display cannot be guaranteed to preserve all pairwise distances of an arbitrary growing dataset; budget for local fidelity only.

When it is useful

• Datasets too large or too high-dimensional for a single trustworthy embedding, where global projections rearrange under small changes.
• Tasks that are local by nature: inspecting a neighbourhood, following a trajectory, comparing near neighbours, auditing a cluster boundary.
• Situations where exact retrieval of the underlying record matters more than the overall picture.
• Exploring spaces that are themselves spherical or have a natural rotation structure, where the lens state and the data share a geometry.

Caveats

• There is no global overview; users can lose their place. Pair the lens with a coarse map or a breadcrumb of visited states.
• Seams, frame singularities, and holonomy are real and will be reported as bugs. They are consequences of the geometry (§§7 and 9); choose an orientation policy and document it.
• A small sphere has little area. Overplotting arrives quickly; the clutter term $\mu E_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}$ and the point-count cap exist for this reason.
• Distortion outside the lens radius is unbounded. Do not read distances between a visible point and one that has just scrolled off.
• Recomputing a layout per frame is expensive. Incremental updates and neighbourhood caching are the difference between a demo and a tool.
• A lensball is not a replacement for global dimensionality reduction. It answers a different question: what does this region look like, and how do I get to the next one.

In mathematics

1. Terminology and notation

For $R>0$, let

$$S_R^n=\{x\in {\mathbb{ℝ}}^{n+1}:|x|=R\}$$

denote the $n$-dimensional sphere of radius $R$. Its intrinsic dimension is $n$, even though it sits in the $(n+1)$-dimensional Euclidean space ${\mathbb{ℝ}}^{n+1}$.

Thus:

• $S^1$ is a circle;
• $S^2$ is the familiar spherical surface in three-dimensional space;
• $S^3$ is the three-dimensional hypersphere in ${\mathbb{ℝ}}^4$.

The closed solid $n$-ball is

$$D_R^n=\{x\in {\mathbb{ℝ}}^n:|x|\le R\}.$$

Unless explicitly stated otherwise, lensball refers to a spherical display surface $S^m$, not a solid ball $D^{m+1}$. The solid-ball version is discussed in §6.3.

Let:

• $A$ be the source space;
• $B$ be the lens space or display space;
• $n=\dim A$ when $A$ is a manifold;
• $m=\dim B$;
• $\theta \in \mathrm{\Theta }$ be the lens state;
• $U_{\theta }\subseteq A$ be the region currently exposed by the lens;
• $F_{\theta }:U_{\theta }\to B$ be the view map;
• $G_{\theta }$ be an inverse, partial inverse, pseudoinverse, or retrieval map when one exists;
• $T$ be a transition rule determining how user motion changes $\theta$.

A lensball may therefore be represented abstractly by a tuple

$$\mathcal{ℒ}=(A,B,\mathrm{\Theta },\{U_{\theta }\},\{F_{\theta }\},\{G_{\theta }\},T).$$

Not every application needs every component. In particular, for finite data, $G_{\theta }$ may simply be an exact dictionary lookup from a displayed point to its original datum.

Three tiers are used below. Lensball is the general mechanism above. A spherical lensball is the canonical geometric case in which $B$ is a sphere and the lens state is a rotation or a choice of subspace (§§2–4 and 10 of the mathematics section); the rotation-group, exponential-map, holonomy, and Grassmannian results are properties of this realization, not of the general notion. A discrete lensball is the finite-data specialization of the discrete lensball definition (§20 of the mathematics section).

1.1 Geometric inverse versus semantic inverse

Two notions of inversion should be distinguished.

A geometric inverse satisfies

$$G_{\theta }(F_{\theta }(x))=x$$

because $F_{\theta }$ itself is injective on its domain.

A semantic inverse may instead store the identity of every displayed datum. Even if a dimensionality-reduction map is not mathematically invertible, selecting a rendered marker can still retrieve the exact original vector, object, or record.

This distinction is important in discrete visualization:

$$\text{lossless identity}\Rightarrow ̸\text{lossless geometry}.$$

A computer can preserve exactly which datum a point represents while necessarily distorting distances, angles, topology, or density in its visual placement.

2. The spherical Lensball: motivating case

The motivating example uses

$$A=S_2^2,B=S_1^2.$$

Thus $A$ is a sphere of radius $2$ and $B$ is a sphere of radius $1$.

Their surface areas are

$$\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(A)=4\pi (2{)}^2=16\pi ,\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(B)=4\pi .$$

Hence $B$ has one quarter of the physical surface area of $A$.

There is a trivial global similarity

$$H:A\to B,H(x)=\frac{x}{2},$$

which is a bijection and scales every intrinsic distance by $1/2$. Therefore the lensball idea is not needed merely to identify the two spheres. The nontrivial idea begins when $B$ is treated as a movable local scope: its full display capacity is devoted to a proper region of $A$, and rotation of the scope changes which region is displayed.

2.1 Equal-area local scope

Suppose one wants one lensball view to represent a spherical cap of $A$ whose area equals the entire area of $B$, namely $4\pi$.

A cap of angular radius $\alpha$ on a sphere of radius $R$ has area

$$A_{\mathrm{c}\mathrm{a}\mathrm{p}}=2\pi R^2(1-\cos \alpha ).$$

For $R=2$,

$$8\pi (1-\cos \alpha )=4\pi ,$$

so

$$\cos \alpha =\frac{1}{2},\alpha =\frac{\pi }{3}=60^{\circ }.$$

Thus an equal-area interpretation of the motivating example corresponds to a cap of angular radius $60^{\circ }$ on $A$.

This is only one possible scale convention. A lensball can instead choose its visible region according to point count, density, geodesic radius, information content, nearest-neighbor structure, or adaptive error.

Figure M2. (Realised: Figure 2 above, segment 1.) Sphere $A=S_2^2$ with a $60^{\circ }$ spherical cap highlighted. The cap has area $4\pi$, equal to the entire area of $B=S_1^2$. The cap is expanded into the lensball view.

3. Local spherical coordinates via the exponential map

A natural smooth implementation uses the Riemannian exponential and logarithm maps.

Let $c\in S_R^n$ be the current lens center. Its tangent space is

$$T_c{S}_R^n=\{v\in {\mathbb{ℝ}}^{n+1}:v\cdot c=0\}.$$

For $w\in T_c{S}_R^n$, with $s=\left|w\right|$, the exponential map on the sphere is

$${\exp }_c(w)=\cos \left(\frac{s}{R}\right)c+R\sin \left(\frac{s}{R}\right)\frac{w}{s},$$

with the limiting value ${\exp }_c(0)=c$.

Conversely, for $p\ne -c$, define

$$\vartheta =\arccos \left(\frac{c\cdot p}{R^2}\right).$$

Then

$${\log }_c(p)=R\frac{\vartheta }{\sin \vartheta }(\frac{p}{R}-\cos \vartheta \frac{c}{R}).$$

where the formula is understood by continuity at $p=c$, giving ${\log }_c(c)=0$. Its norm is the geodesic distance:

$$|{\log }_c(p)|=R\vartheta .$$

The antipodal point $-c$ is the cut locus of $c$ on a sphere: the shortest geodesic from $c$ ceases to be unique there, so the logarithm map cannot be made single-valued at that point.

3.1 A local Lensball map between equal-dimensional spheres

Let

$$A=S_R^n,B=S_r^n.$$

Choose a display center $b_0\in B$ and an orthogonal identification of tangent spaces

$$P:T_{c}A\to T_{b_0}B.$$

For an angular magnification factor $\eta >0$, define

$$F_{c,P,\eta }(p)={\exp }_{b_0}^B(\eta \frac{r}{R}P{\log }_c^A(p)).$$

If $p$ lies at angular distance $\vartheta$ from $c$, its image lies at angular distance $\eta \vartheta$ from $b_0$.

On a sufficiently small cap this map is one-to-one and has inverse

$$G_{c,P,\eta }(b)={\exp }_c^A(\frac{R}{\eta r}{P}^{-1}{\log }_{b_0}^B(b)).$$

The factor $\eta$ plays the role of zoom. When $\eta >1$, a small region of $A$ is expanded over a larger portion of $B$. The domain must be kept away from the relevant cut loci; for example, one can require

$$\vartheta <\min (\pi ,\frac{\pi }{\eta }).$$

For the special case of two complete spheres of the same dimension and proportional radii, the global radial similarity is exact. The local exponential-map formulation is useful because it generalizes to arbitrary manifolds and to adaptive local magnification.

4. Navigation and rotation

For the ordinary spherical case $A=S_R^2$, a complete oriented lensball state can be represented by a rotation matrix

$$Q_t\in SO(3).$$

If $e_3=(0,0,1{)}^T$, the current center may be represented as

$$c_t=R{Q}_t{e}_3.$$

The matrix $Q_t$ contains more information than the center $c_t$ alone. A point on $S^2$ has two degrees of freedom, whereas a full oriented local frame has three. The third degree of freedom determines the lensball's local notion of "up."

Suppose a user rotates $B$ by an incremental rotation

$$\mathrm{\Delta }R=\exp (\mathrm{\Omega }),\mathrm{\Omega }\in \mathfrak{s}\mathfrak{o}(3),$$

where $\mathfrak{s}\mathfrak{o}(3)$ is the Lie algebra of skew-symmetric $3\times 3$ matrices. A basic sensitivity rule is

$$Q_{t+1}=Q_t\exp (\kappa \mathrm{\Omega }),$$

where $\kappa >0$ is a navigation sensitivity.

• $\kappa =1$: equal angular response;
• $0<\kappa <1$: fine motion;
• $\kappa >1$: accelerated traversal.

For $A=S_R^2$ and $B=S_r^2$, equal angular motion and equal arc-length motion are different:

$$s=R\vartheta .$$

Equal angular motion uses the same $\vartheta$. Equal physical arc length requires

$$R{\vartheta }_A=r{\vartheta }_B,{\vartheta }_A=\frac{r}{R}{\vartheta }_B.$$

For the motivating radii $R=2$, $r=1$, equal arc-length sensitivity gives

$${\vartheta }_A=\frac{1}{2}{\vartheta }_B.$$

5. Dimensional cases

The qualitative behavior of a lensball depends strongly on the relation between

$$n=\dim A\text{and}m=\dim B.$$

Case	Interpretation	Generic consequence
$n=m$	local chart or local reparameterization	locally invertible maps are possible
$n>m$	projection, slice, quotient, or fibered view	some geometric information must be hidden
$n<m$	embedding into a higher-dimensional view	extra display dimensions are available
finite/discrete $A$	exact identities plus approximate geometry	point identity may be lossless even when geometry is not

5.1 Equal dimension: $n=m$

If $A$ and $B$ are smooth manifolds of the same dimension, then sufficiently small neighborhoods are locally equivalent as smooth manifolds. In particular, after choosing charts,

$$U\subset A⟷V\subset {\mathbb{ℝ}}^n⟷W\subset B.$$

Thus a lensball can be locally one-to-one and smoothly invertible.

If $A$ and $B$ carry Riemannian metrics, their local metrics need not agree exactly, but their distortion can be made arbitrarily small on sufficiently small regions after choosing an appropriate first-order scaling.

This is the mathematical basis for the claim that local geometric error can be made arbitrarily small in the equal-dimensional smooth case.

5.2 Source dimension larger than lens dimension: $n>m$

When $n>m$, a lensball cannot continuously and injectively represent an open $n$-dimensional neighborhood inside an $m$-dimensional manifold. Something must be discarded, merged, or selected.

The principal options are:

Projection

All points are mapped to a lower-dimensional representation,

$$F:A\to B.$$

Different source points may share the same image.

Slice

The lensball displays an $m$-dimensional subspace or submanifold of the source.

For example, if

$$A=S^{d-1}\subset {\mathbb{ℝ}}^d$$

and $V\subset {\mathbb{ℝ}}^d$ is an $(m+1)$-dimensional linear subspace, then

$$A\cap V\cong S^m.$$

This gives an exact spherical $m$-dimensional slice.

Thick slice

For discrete data, exact intersection with a slice is often empty or sparse. One therefore selects points near a slice,

$$X_V(\delta )=\{x\in X:\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,V)<\delta \},$$

and maps those points onto $B$.

Quotient or fibered representation

A map

$$F:A\to B$$

groups source points into fibers

$$F^{-1}(b)=\{a\in A:F(a)=b\}.$$

A single lensball position then represents an entire hidden set in $A$.

The classical Hopf fibration

$$S^3\to S^2$$

is a canonical mathematical example: every point of $S^2$ corresponds to a circle $S^1$ in $S^3$. It is not a generic data-visualization method, but it is an instructive model of how a lower-dimensional spherical representation can systematically encode higher-dimensional structure through fibers.

Figure M3 (proposed). Dimension mismatch. Several points or a continuous fiber in high-dimensional $A$ map to one location on $B$. A second panel shows a moving $S^2$ slice through a higher-dimensional sphere.

5.3 Source dimension smaller than lens dimension: $n<m$

When $n<m$, local embeddings are possible and the lensball has excess geometric capacity.

The simplest example is the equatorial embedding

$$S^n\hookrightarrow S^m,n<m.$$

For a general smooth $n$-manifold, the Whitney embedding theorem guarantees a smooth embedding into ${\mathbb{ℝ}}^{2n}$. Consequently, sufficiently high-dimensional lensball spaces can globally embed arbitrary smooth manifolds, although a particular manifold may embed in substantially lower dimension.

Extra dimensions can also encode non-geometric variables such as uncertainty, class, time, density, velocity, or provenance.

6. Two-dimensional and three-dimensional Lensballs

The word "dimension" can be misleading because a visible sphere in ordinary three-dimensional graphics is intrinsically two-dimensional.

6.1 One-dimensional Lensball

The simplest toy model is

$$B=S^1.$$

A local segment of a curve can be wrapped around a circle. Moving the circle's state along the source curve gives a one-dimensional analogue of lensball navigation.

This case is useful for illustrating the difference between a closed display $S^1$ and a local interval. A whole circle cannot be homeomorphic to an interval without a cut, overlap, or identification.

6.2 Two-dimensional Lensball

The familiar visual form is

$$B=S^2.$$

It is a two-dimensional surface rendered naturally in three-dimensional computer graphics. This is the main practical lensball for human visual inspection.

A two-dimensional lensball can display:

• a local surface patch of a two-dimensional manifold;
• a projection of arbitrary-dimensional data;
• a two-dimensional slice of a higher-dimensional manifold;
• or a finite set of high-dimensional data points embedded on a sphere.

6.3 Three-dimensional Lensball

An intrinsically three-dimensional spherical lensball is

$$B=S^3\subset {\mathbb{ℝ}}^4.$$

It cannot be viewed directly in ordinary three-dimensional space without another projection or slicing step.

A different practical object is the volumetric lensball

$$B=D^3,$$

a solid three-dimensional ball. It can use the interior as display space. This increases display dimension from two to three but introduces familiar problems of occlusion and interior navigation.

Thus a "3D ball" in graphics and a "3-sphere" in topology are distinct objects:

$$D^3\subset {\mathbb{ℝ}}^3\text{versus}{S}^3\subset {\mathbb{ℝ}}^4.$$

7. Topological perspective

Lensball has a natural interpretation in terms of charts, atlases, embeddings, and quotient maps.

7.1 Lensball as a moving chart

An $n$-dimensional manifold is locally modeled on ${\mathbb{ℝ}}^n$. A chart is a homeomorphism

$$\phi :U\to V\subset {\mathbb{ℝ}}^n.$$

A collection of compatible charts covering a manifold is an atlas.

An equal-dimensional lensball is essentially a stateful chart whose target is another manifold rather than necessarily ${\mathbb{ℝ}}^n$. Moving the lensball changes the chart domain $U_{\theta }$. The collection of all reachable lens states behaves like a navigable atlas.

This viewpoint explains why local representation is easy while global representation may require multiple charts, cuts, or overlaps.

7.2 Why a whole sphere is not a local disk

A proper local patch of an $n$-manifold is ordinarily disk-like. The sphere $S^n$ is not homeomorphic to the disk $D^n$.

One proof uses homology:

$$H_n(S^n)\cong \mathbb{\mathbb{Z}},H_n(D^n)=0.$$

Homeomorphisms preserve homology groups, so no homeomorphism

$$S^n\cong D^n$$

can exist for $n\ge 1$.

Therefore, if the entire lensball surface $S^n$ is required to correspond continuously and bijectively to one disk-like local patch of $A$, some seam, removed point, overlap, or identification is unavoidable.

This is not a practical problem if the lensball is treated as a display shell rather than as a literal topological chart: one can reserve a seam, use overlapping charts, or keep the singular region outside the active visual field.

7.3 Invariance of domain and invariance of dimension

Brouwer's invariance of domain theorem states that if

$$U\subset {\mathbb{ℝ}}^n$$

is open and

$$f:U\to {\mathbb{ℝ}}^n$$

is continuous and injective, then $f(U)$ is open and $f$ is a homeomorphism onto its image.

It also gives a short obstruction when $n>m$.

Suppose there were a continuous injective map

$$f:U\subset {\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m,m<n.$$

Embed ${\mathbb{ℝ}}^m$ into ${\mathbb{ℝ}}^n$ as the coordinate plane

$$i(y_1,\dots ,y_m)=(y_1,\dots ,y_m,0,\dots ,0).$$

Then

$$i\circ f:U\to {\mathbb{ℝ}}^n$$

would be continuous and injective. Invariance of domain would imply that $(i\circ f)(U)$ is open in ${\mathbb{ℝ}}^n$. But it lies entirely in an $m$-dimensional coordinate plane, which has empty interior in ${\mathbb{ℝ}}^n$. Contradiction.

Hence an open $n$-dimensional region cannot be continuously injected into an $m$-dimensional display when $m<n$.

This is one of the fundamental limits of a lower-dimensional lensball.

7.4 Borsuk–Ulam theorem

The Borsuk–Ulam theorem states that every continuous map

$$f:S^n\to {\mathbb{ℝ}}^n$$

identifies at least one antipodal pair:

$$f(x)=f(-x)$$

for some $x\in S^n$.

Invariance of domain (§7.3) already shows that a continuous global flattening of $S^n$ into an $n$-dimensional Euclidean display cannot be injective; Borsuk–Ulam sharpens this by locating a specific collision, an antipodal pair. It does not forbid a sphere-to-sphere bijection $S^n\to S^n$; the obstruction is specific to a flat $n$-dimensional target.

7.5 Hairy ball theorem and lens orientation

The hairy ball theorem says that an even-dimensional sphere $S^{2k}$, in particular $S^2$, has no continuous nowhere-zero tangent vector field.

For an $S^2$ lensball this has a practical interpretation: one cannot choose a globally smooth, non-singular "up direction" that depends only on the center point of the lens.

There are two standard solutions:

1. accept coordinate singularities, as with longitude/latitude; or
2. include orientation as part of the lens state.

The second solution motivates representing the state by $Q\in SO(3)$, not merely by the center $c\in S^2$.

8. Differential geometry: local fidelity and curvature

8.1 Arbitrarily small local distortion

Let $A$ and $B$ be smooth $n$-dimensional Riemannian manifolds, and choose points

$$p\in A,q\in B.$$

Choose a linear isometry, or a scaled isometry,

$$J:T_{p}A\to T_{q}B.$$

Using exponential maps, define locally

$$F={\exp }_q\circ J\circ {\log }_p.$$

For sufficiently small neighborhoods this is a diffeomorphism.

In Riemannian normal coordinates centered at a point, the metric has expansion

$$g_{ij}(x)={\delta }_{ij}-\frac{1}{3}{R}_{ikjl}(p)x^k{x}^l+O(|x{|}^3),$$

where $R_{ikjl}$ is the Riemann curvature tensor.

The first deviation from the Euclidean metric is therefore quadratic in distance. Consequently, on a region of characteristic diameter $L$ and curvature scale $K$,

$$\text{local metric distortion}=O(K{L}^2).$$

For a sphere of radius $R$,

$$K=\frac{1}{R^2},$$

giving the heuristic scaling

$$\text{error}=O\left(\frac{L^2}{R^2}\right).$$

Thus, in the equal-dimensional smooth setting, shrinking the lens region can make local metric distortion arbitrarily small.

Proof sketch

At the center point, normal coordinates make the metric exactly Euclidean:

$$g_{ij}(0)={\delta }_{ij},$$

and the first derivatives of the metric vanish:

$${\partial }_k{g}_{ij}(0)=0.$$

The metric therefore converges uniformly to its tangent-space metric as the neighborhood radius tends to zero. The same holds in $B$. Composing the two normal-coordinate systems through $J$ yields a map whose differential approaches an isometry uniformly as the domain shrinks. Hence, for every $\epsilon >0$, a sufficiently small neighborhood is $(1+\epsilon )$-bi-Lipschitz after the chosen scale normalization.

This statement is local. It does not imply that global curvature or topology can be made irrelevant while viewing the entire manifold at once.

8.2 Cost of reducing local error

For a two-dimensional surface, if linear patch size is reduced by a factor $a$, its area is reduced roughly by $a^2$. Since curvature error is also second order,

$$\epsilon \sim L^2,$$

one expects, very roughly,

$$N_{\text{views}}\sim \frac{1}{\epsilon }$$

to cover a fixed two-dimensional surface at uniformly comparable local accuracy.

In $n$ intrinsic dimensions, a patch volume scales like $L^n$. If distortion remains quadratic in $L$,

$$N_{\text{views}}\sim L^{-n}\sim {\epsilon }^{-n/2}.$$

This is a geometric form of the curse of dimensionality: arbitrary local accuracy remains possible, but the number of neighborhoods needed for uniform coverage may grow rapidly with dimension.

9. Parallel transport, holonomy, and geometric memory

Suppose the lensball carries an oriented tangent frame and is moved along a path on a curved source manifold.

A natural way to preserve orientation is parallel transport: move the frame while keeping it as parallel as the Levi-Civita connection allows.

On a curved manifold, parallel transport around a closed loop need not return the frame to its original orientation. This effect is called holonomy.

For a sphere of radius $R$, parallel transport around a simple loop enclosing region $D$ produces, up to orientation and modulo $2\pi$, a rotation whose angle is

$$\mathrm{\Delta }\varphi ={\int }_{D}KdA=\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(D)}{R^2}.$$

This is closely related to the Gauss–Bonnet theorem.

Lensball therefore has a natural path-dependent phenomenon:

moving the lens around a closed route can return it to the same center with a different orientation.

This is not numerical error; it is a genuine consequence of curvature.

It also creates a design choice. A lensball implementation may:

• preserve orientation by parallel transport and expose holonomy;
• continuously correct orientation against a global reference, introducing singularities or gauge choices;
• or treat orientation as an independent user-controlled state.

Figure M4. (Realised: Figure 2 above, segment 3.) A lensball frame parallel-transported around a triangular loop on $S^2$. The center returns to its starting point but the local frame is rotated. The enclosed area determines the holonomy angle.

10. Higher-dimensional spherical sources and Grassmannians

For high-dimensional spheres, a particularly clean lensball model uses moving great subspheres.

Let

$$A=S^{d-1}\subset {\mathbb{ℝ}}^d.$$

To display an $m$-sphere, choose an $(m+1)$-dimensional linear subspace

$$V\subset {\mathbb{ℝ}}^d.$$

Then

$$A\cap V\cong S^m.$$

Thus the lensball state can be taken to be the choice of $V$.

The set of all $(m+1)$-dimensional linear subspaces of ${\mathbb{ℝ}}^d$ is the Grassmannian

$$\mathrm{G}\mathrm{r}(m+1,d).$$

Its dimension is

$$\dim \mathrm{G}\mathrm{r}(m+1,d)=(m+1)(d-m-1).$$

This gives an immediate quantitative description of the space of possible spherical slices.

10.1 Example: $S^4$ viewed through $S^2$

Take

$$A=S^4\subset {\mathbb{ℝ}}^5,B=S^2.$$

A great $S^2$ slice is determined by a 3-dimensional subspace of ${\mathbb{ℝ}}^5$:

$$V\in \mathrm{G}\mathrm{r}(3,5).$$

The state space has dimension

$$3(5-3)=6.$$

By contrast, the visible ball $S^2$ has rotation group

$$SO(3),$$

which has dimension

$$\frac{3(3-1)}{2}=3.$$

Thus instantaneous rotation of the visible ball supplies only three control degrees of freedom, while the family of possible slices has six dimensions.

This does not imply that the full slice space is unreachable. It means that a transition rule must determine how low-dimensional controls accumulate over time to move through the larger state space.

Figure M5 (proposed). An $S^2$ lensball controlling a moving 3-plane $V$ through ${\mathbb{ℝ}}^5$. Each plane cuts $S^4$ in a visible $S^2$. The sequence of planes traces a path in $\mathrm{G}\mathrm{r}(3,5)$.

11. Control-theoretic perspective

When the lens state $\theta$ lies on a manifold $\mathrm{\Theta }$, user input can be modeled as a control system

$$\dot{\theta }=\sum _{i=1}^k{u}_i(t)X_i(\theta ),$$

where:

• $u_i(t)$ are control signals derived from lensball motion;
• $X_i$ are vector fields on the lens-state space.

This formulation is useful when the number $k$ of instantaneous controls is smaller than

$$\dim \mathrm{\Theta }.$$

A familiar analogy is a car: it has fewer instantaneous control directions than the dimension of its configuration space, yet sequences of steering and forward/backward motion can reach positions that are not instantaneously accessible.

The relevant mathematical principle is the Chow–Rashevskii theorem. Informally, if the control vector fields and their iterated Lie brackets span the full tangent space,

$$\mathrm{L}\mathrm{i}\mathrm{e}\{X_1,\dots ,X_k\}(\theta )=T_{\theta }\mathrm{\Theta },$$

then the system is controllable: any state can be reached from any other by admissible controls. The hypotheses are that the $X_i$ are smooth on a connected manifold, the Lie algebra they generate spans the tangent space at every point, and the controls are unconstrained in sign (Agrachev and Sachkov, ch. 5).

For the canonical oriented $S^2\to S^2$ rotational realisation, the rotational state manifold is ${\mathrm{\Theta }}_{\mathrm{r}\mathrm{o}\mathrm{t}}=SO(3)$ and the natural controls are the left-invariant fields $X_i(Q)=Q{\hat{e}}_i$, $i=1,2,3$, where ${\hat{e}}_i$ is the skew matrix of the axis $e_i$. With all three controls the system is trivially controllable. With only two, say tilts about $e_1$ and $e_2$, the bracket $[X_1,X_2]=X_3$ supplies the missing direction, so the condition holds and every orientation is reachable by sequences of two tilts. This is why a two-axis input such as a mouse drag can steer a three-dimensional frame, provided the input is actually mapped to two signed body-fixed rotations or another bracket-generating pair; an arbitrary two-axis mapping is not automatically controllable. When the transition rule lifts $SO(3)$ motion to a larger state space (a Grassmannian, §10), controllability must be re-checked for the lifted fields; it is not inherited.

This makes the transition rule more than a sensitivity constant. It can itself be a geometric navigation mechanism.

12. Discrete and computational Lensballs

For a finite dataset

$$X=\{x_1,\dots ,x_N\}\subset A,$$

the computational problem differs sharply from the continuous topological problem.

At state $\theta$, let

$$X_{\theta }\subseteq X$$

be the currently visible subset, and let

$$f_{\theta }:X_{\theta }\to B$$

assign display positions.

12.1 Exact identity is easy

If the renderer has enough distinguishable markers, $f_{\theta }$ can be injective on a finite visible set. Each rendered marker can store an identifier $i$, so selecting the marker recovers $x_i$ exactly.

Thus the round trip

$$x_i⟶f_{\theta }(x_i)⟶i⟶x_i$$

can be computationally exact even if the geometric placement is highly distorted.

For a finite set, this eliminates the topological obstruction to identity preservation. It does not eliminate geometric obstruction.

12.2 Global metric preservation remains limited

Suppose $X\subset {\mathbb{ℝ}}^d$ contains $N$ points and the goal is to preserve every pairwise Euclidean distance within relative error $\epsilon$.

The Johnson–Lindenstrauss lemma states, roughly, that one can embed the points into

$${\mathbb{ℝ}}^k$$

with

$$k=O\left(\frac{\log N}{{\epsilon }^2}\right)$$

while preserving all pairwise distances up to a factor $1\pm \epsilon$.

The bound is also tight in the worst case: for some point sets any map, linear or not, with distortion $1\pm \epsilon$ needs $k=\mathrm{\Omega }({\epsilon }^{-2}\log N)$ (Larsen and Nelson, 2017). Hence a fixed two-dimensional or three-dimensional display cannot, for arbitrary growing datasets, preserve all high-dimensional pairwise geometry with arbitrarily small error.

Discrete computation makes exact point identity possible; it does not repeal dimensional geometry.

Lensball addresses this by changing the objective from global simultaneous fidelity to local, adaptive, and temporal fidelity.

13. Fidelity objectives for data visualization

A practical lensball needs a definition of what should be preserved.

For visible points $x_i,x_j\in X_{\theta }$, let

$$d_A(x_i,x_j)$$

be the relevant source-space distance and

$$d_B(f_{\theta }(x_i),f_{\theta }(x_j))$$

the display distance.

One possible local geometric stress is

$$E_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(\theta )={\sum }_{(i,j)\in E_{\theta }}{w}_{ij}{(d_B(f_{\theta }(x_i),f_{\theta }(x_j))-s_{\theta }{d}_A(x_i,x_j))}^2,$$

where:

• $E_{\theta }$ is a selected neighborhood graph;
• $w_{ij}$ emphasizes important pairs;
• $s_{\theta }$ is a local display scale.

To prevent points from jumping arbitrarily between consecutive views, define a temporal penalty

$$E_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}({\theta }_t,{\theta }_{t-1})={\sum }_{x_i\in X_{{\theta }_t}\cap X_{{\theta }_{t-1}}}{d}_B{(f_{{\theta }_t}(x_i),f_{{\theta }_{t-1}}(x_i))}^2.$$

A simple lensball objective is then

$$E=E_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}+\lambda E_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}+\mu E_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}+\nu E_{\mathrm{t}\mathrm{a}\mathrm{s}\mathrm{k}},$$

where further terms may penalize visual collisions or reward a task-specific property.

This formulation separates two requirements that are often conflated:

• spatial fidelity: the current picture is geometrically useful;
• temporal fidelity: motion through pictures is cognitively continuous.

For interactive high-dimensional visualization, temporal fidelity may be as important as the static embedding quality.

This form is standard. The local stress on a neighbourhood graph belongs to the local-MDS and neighbourhood-preservation family, for which Venna and Kaski (2006) give the standard trustworthiness–continuity framing that guides the choice of $E_{\theta }$ and $w_{ij}$; the temporal penalty is the same quadratic consecutive-layout regularisation pattern used by Xu, Kliger and Hero (2012). What lensball fixes is only that both terms are evaluated along the trajectory induced by the transition rule, so $\lambda$ trades off against the declared navigation law rather than against an arbitrary sequence of snapshots.

Figure M6 (proposed). The same local point set in two consecutive lensball states. A stable transition moves shared points slightly; an unstable recomputation preserves similar static stress but permutes the visual arrangement dramatically.

14. Lensball as projection, slice, and neighborhood

Three lensball modes should be distinguished.

Projection Lensball

A large or complete source set is mapped into $B$:

$$F_{\theta }:A\to B.$$

The state changes the projection itself. Classical projection tours fit naturally into this category.

Slice Lensball

The state selects a lower-dimensional subset

$$M_{\theta }\subset A$$

and maps only that slice into $B$.

For spherical sources, Grassmannian slices provide a canonical construction.

Neighborhood Lensball

The state selects a center or anchor $c_{\theta }$ and a local neighborhood

$$U_{\theta }=\{x\in A:d_A(x,c_{\theta })<{\rho }_{\theta }\}.$$

The visible geometry is then recomputed locally.

These modes can be combined. For example, a system may select a local neighborhood and then project it onto a state-dependent spherical slice.

15. Other mathematical perspectives

15.1 Fiber bundles

When $n>m$, the inverse image

$$F^{-1}(b)$$

of a lensball point may have positive dimension. If the fibers vary continuously and locally have a common structure, the map may resemble or form a fiber bundle.

This gives a precise language for "hidden dimensions behind a displayed point."

The Hopf fibration of §5.2 is the standard spherical example.

15.2 Lie groups and homogeneous spaces

Rotations of $S^n$ are described by

$$SO(n+1).$$

The sphere itself can be written as a homogeneous space

$$S^n\cong SO(n+1)/SO(n).$$

This explains the difference between a lens center and a fully oriented lens frame. The center forgets the $SO(n)$ rotation of the tangent frame.

Similarly, Grassmannians are homogeneous spaces:

$$\mathrm{G}\mathrm{r}(k,d)\cong O(d)/(O(k)\times O(d-k))$$

in the unoriented real case.

This makes Lie-group integration, exponential maps, and geodesic interpolation natural tools for smooth lensball navigation.

15.3 Gauge viewpoint

Choosing a local orientation or coordinate frame over every point of a manifold is analogous to choosing a gauge. Different frame conventions can represent the same underlying geometric state.

Seams, frame singularities, and holonomy are therefore not merely graphics problems; they are manifestations of the geometry of bundles and connections.

15.4 Information and rate-distortion viewpoint

A low-dimensional visual display has finite perceptual and pixel capacity. From an information-theoretic viewpoint, the lensball chooses which information to preserve at each state.

Instead of demanding a globally sufficient static encoding, lensball can be regarded as a progressive interactive code:

$$\text{space}⟶{\text{local view}}_1,{\text{local view}}_2,\dots$$

with user motion selecting the next packet of information.

The data itself need not be compressed destructively; only the momentary visual channel is.

15.5 Numerical-analysis viewpoint

An implementation based on local charts resembles adaptive numerical methods:

• decrease lens radius where curvature or data complexity is high;
• increase radius in nearly flat or homogeneous regions;
• refine sampling where local error exceeds tolerance;
• reuse previous solutions to obtain temporal stability.

This suggests adaptive lensball schemes whose visible radius is determined by an error estimator rather than by a fixed angle.

16. Relevant theorems and their Lensball interpretations

Result	Lensball relevance
Local Euclidean structure of manifolds	equal-dimensional neighborhoods admit local coordinates and local inverses
Riemannian normal-coordinate expansion	local metric error is second order in neighborhood size
Invariance of domain	an open $n$-region cannot be continuously injected into lower dimension
Borsuk–Ulam theorem	a continuous flattening $S^n\to {\mathbb{ℝ}}^n$ must identify an antipodal pair
Whitney embedding theorem (1944)	every smooth $n$-manifold embeds in ${\mathbb{ℝ}}^{2n}$; a display of dimension $2n$ always suffices for an injective picture, though not an isometric one
Hairy ball theorem	a global non-singular tangent "up" direction on $S^2$ is impossible
Gauss–Bonnet theorem / holonomy	transporting a lensball frame around curvature can rotate its orientation
Johnson–Lindenstrauss lemma and its lower bound	$N$ points admit a $(1\pm \epsilon )$-distance-preserving map into dimension $O({\epsilon }^{-2}\log N)$; there exist point sets for which this is necessary for any map, linear or not (Larsen–Nelson), so a fixed low-dimensional lens cannot be guaranteed uniform fidelity on arbitrary large sets, and locality is what buys accuracy back
Chow–Rashevskii theorem	low-dimensional controls can navigate a higher-dimensional state manifold if the induced vector fields are bracket-generating; on $SO(3)$ two tilts suffice (§11)

No single theorem determines whether a lensball visualization is useful: topology decides whether exact continuous representations are possible, differential geometry quantifies local distortion, metric embedding theory constrains distance preservation, control theory determines reachability, and perception adds constraints none of them see. Lensball is a framework connecting these questions, not a projection theorem.

17. Relation to existing work

Interactive lenses (parent category)

Bier et al.'s Magic Lenses (1993) introduced a movable screen region with an operator that changes how the objects beneath it are shown. Tominski et al. (2017) give the selection, lens-function, join model used under Positioning and a taxonomy of more than fifty lens techniques. Lensball is a member of this family with selection at the data stage and a separate spherical view. Among the systems examined in the 2026-09-14 audit, none was identified in which rotation of a spherical rendered view is itself mapped by an explicit transition law to navigation through a source-state manifold.

Movable-lens re-embedding

TopicLens (Kim et al., 2017) drags a rectangular lens over a global document embedding and recomputes a topic model and a local t-SNE for the captured documents, anchoring local topic centroids to their global positions. General Projective Maps (Lehmann and Theisel, 2016) define a data-dependent magic lens that re-solves the projection's free parameters by least squares so that points inside the lens spread apart. Focus+Context Exploration of Hierarchical Embeddings (Höllt et al., 2019) formalises eight focus and context operations as set equations on a tree-shaped state and extends HSNE so that mixed levels of detail embed together. TopicLens and General Projective Maps recompute or re-optimise in response to a movable planar region and stabilise against the global layout; Höllt et al. instead define explicit focus and context state operations and recompute hierarchical embeddings, warm-starting from prior positions. All three are planar. Lensball takes from them the local-recompute pattern, the identity link from displayed point to source datum, and (from Höllt et al.) the idea that interaction operations belong to the model; it adds a continuous geometric transition law and a continuity objective evaluated along the trajectory that law induces.

Spherical embedding and spherical displays

Nonmetric spherical MDS (Cox and Cox, 1991); Visualizing Information on a Sphere (Gross, Sprenger and Finger, 1997); the Spherical Similarity Explorer (Zhang et al., 2016), which already discusses moving the point of interest while preserving the analyst's mental map; DOSNES (Lu, Corander and Yang, 2019), which shows that doubly stochastic similarities embed naturally on spheres; and scPhere (Ding and Regev, 2021). These establish spherical targets. Lensball uses the sphere as a moving local chart, not as a global target.

Focus+context on spheres and hemispheres

Generalized fisheye views (Furnas, 1986); MagicSphere (Cignoni, Montani and Scopigno, 1994), a movable spherical volume of interest in 3D; the Magic Eye View (Kreuseler and Schumann, 1999; 2002), a hierarchy projected onto a hemisphere with a movable focus; iSphere (Du et al., 2017), a graph mapped to a Riemann sphere for focus+context; and Sphere (Benko, Wilson and Balakrishnan, 2008), a multi-touch spherical display whose rotation is the navigation. These are interactive spherical representations already. The distinction is the source-state coupling: in a lensball the sphere is a chart of a source manifold, and its rotation is lifted by a declared law to a state that may be larger than the rotation group.

Tours, slices, and local projection

Asimov's grand tour (1985) and its guided descendants (Buja, Cook, Asimov and Hurley, 2005) move through projection planes along geodesics of the Grassmannian; a projection lensball whose state is a subspace is a state-controlled relative. Lifting rotation to a data graph or a general manifold is a different problem, closer to path planning than to tours. HyperSlice (van Wijk and van Liere, 1993) explores a high-dimensional function through interactive lower-dimensional slices, the ancestor of the slice lensball. Charting a Manifold (Brand, 2002) and LAMP (Joia et al., 2011) supply the local patch and local affine machinery that a neighbourhood lensball can use.

Local fidelity and temporal regularisation

Local MDS (Venna and Kaski, 2006) makes the trustworthiness–continuity trade-off explicit for local proximities; regularised dynamic layout (Xu, Kliger and Hero, 2012) adds a temporal movement penalty to a static layout objective. §13 is these two objectives evaluated along a lensball trajectory.

18. Design principles

A computational lensball can be summarized by the following principles.

Preserve identity exactly when possible

The visual position is a view, not the source datum. Every rendered point should retain its original identifier and coordinates.

Preserve local geometry according to the task

There is no universal best metric. Depending on the application, a lensball may prioritize:

• geodesic distance;
• Euclidean distance;
• nearest-neighbor rank;
• angles;
• cluster membership;
• density;
• topology;
• class separation;
• uncertainty.

Make the transition rule explicit

Movement of the lensball is itself part of the mathematical model. A system should specify how control input changes:

• lens center;
• lens orientation;
• selected subspace;
• zoom;
• slice thickness;
• embedding parameters.

Optimize continuity over time

Two individually good embeddings can form a bad interactive system if one abruptly rearranges the other.

The trajectory

$$t\mapsto F_{\theta (t)}$$

is therefore a primary object, not an incidental animation between static plots.

Treat global impossibility as a navigation problem

If a low-dimensional display cannot represent all of $A$ faithfully at once, lensball does not attempt to defeat the theorem. It changes the interface:

$$\text{global high-dimensional object}⇝\text{sequence of faithful local experiences}.$$

19. Open and settled questions

Most questions a lensball raises are specialisations of named problems. The list below says which, and what is still open once the lensball data (source metric, transition rule, task) are fixed.

1. Local map. Choosing $F_{\theta }$ to minimise local distortion is spherical or local multidimensional scaling once a distortion functional and neighbourhood graph are declared (Cox and Cox; Venna and Kaski). Open: which functional matches which task, and whether the equal-area full-display profile of §2.1 is optimal for any natural criterion.
2. Navigation law and controllability. On $SO(3)$ two tilt controls are bracket-generating (§11), so reachability is settled. For Grassmannian projection states this connects to projection-tour navigation (Buja, Cook, Asimov and Hurley); for data graphs or general manifolds it is a path-planning problem. In either case no universal optimum exists because the criterion (coverage, shortest path, information gain, predictability) is task-dependent. Open: intuitiveness of a given lift, an empirical question.
3. Frame transport. A globally smooth "up" field on $S^2$ is impossible (hairy ball), so the choice is between parallel transport of the tangent frame (the directly applicable construction; the rotation-minimising frames of computer graphics are the analogous minimum-twist device), a global reference with seams, or orientation carried as extra state. Holonomy is then determined by Gauss–Bonnet (§9). Settled mathematically; the design choice and its perceptual cost are open.
4. Adaptive scale. Choosing the lens radius from curvature, sample density, or embedding error is adaptive-neighbourhood or bandwidth selection. Open: a scale rule that is stable under the transition rule, so that zoom does not fight navigation.
5. Discrete guarantees. For finite data, uniform pairwise fidelity in a fixed low dimension is impossible in the worst case (Larsen–Nelson lower bound, §12.2). Tracking births and deaths of visible topological features along a trajectory is zigzag persistence. Open: rank-order or neighbourhood guarantees that hold within a moving lens of bounded size, as a function of lens radius and local intrinsic dimension.
6. Perceptual invariants. Which quantities must stay stable across a trajectory for an observer to keep a reliable mental model. Genuinely open and empirical. Among the works reviewed in §17, no study was identified that isolates which invariants across successive re-embedded lens states determine mental-map stability.

20. Discrete Lensball

For a finite dataset, the definition in the lead can be strengthened:

Discrete lensball. A discrete lensball is a lensball in which displayed points retain exact references to their original data objects; geometric distortion affects the view but need not alter the stored data or point identity.

Positioning

Lensball is a synthesis and a name, not a new theorem or a new dimensionality-reduction algorithm. It does not claim new dimensionality reduction, new lens mathematics, new temporal regularisation, or new geometric control theory.

In plain terms: interactive lenses are a well-studied family of tools in which a movable region selects part of a picture and shows it differently (Bier et al., 1993; Tominski et al., 2017). A lensball is one of them. Its lens selects part of the data (a neighbourhood, slice, or projection of $A$) rather than part of a picture, and the result is drawn on a separate spherical view. In the vocabulary of Tominski et al., the selection occurs at the data stage and the lens representation is rendered as a separate view.

The proposed contribution is a particular combination, not any single ingredient:

1. the lens state evolves by an explicit geometric, continuous law (the transition rule), declared as part of the model;
2. temporal fidelity is evaluated along the trajectory that this law induces from the user's motion, not between arbitrary snapshots;
3. rotating the displayed sphere is mapped by that law to navigation through the source's neighbourhoods, slices, or projections, so the sphere's own geometry constrains navigation: the point opposite the lens centre is a seam (cut locus), and a round trip on a curved source can bring the view back rotated (holonomy).

Each ingredient has precedents: explicit state-changing operations (Höllt et al., 2019), temporal penalties on layouts (Xu, Kliger and Hero, 2012), interactive spherical displays (Benko, Wilson and Balakrishnan, 2008; iSphere). The audit of 2026-09-14 did not identify their exact combination in the systems it reviewed. Lensball is not claimed to be the first movable-lens re-embedding interface (TopicLens; General Projective Maps), the first spherical dimensionality reduction (Cox and Cox, 1991), the first spherical information display (Gross, Sprenger and Finger, 1997), or the first focus+context sphere (Magic Eye View; iSphere). See §17.

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Notes on scope

See Positioning: the term names a synthesis whose ingredients are established subjects, and §17 lists the closest prior work.

heidict entry

Lensball — /ˈlɛnz.bɔːl/ · noun · sentiment: neutral.

Inflections

Form	Value
plural	lensballs	possessive	lensball's

Definitions

1. (technical) — A small movable display, usually a sphere, that shows a local view of a much larger source space, together with an explicit rule for how moving the lens changes the view. Turn the ball and the view slides across the source; every displayed point keeps its identity, nothing jumps, and the transition rule is part of the object rather than an afterthought. Three tiers are used: the general mechanism (any lens space, any source), the spherical lensball (lens is a round sphere, state is a rotation), and the discrete lensball (a finite point set with exact identity). A scope, not a map: it gives up one globally faithful picture in exchange for local fidelity and continuity under motion.
   • Give me a lensball over the embedding, not another flat scatter plot.
   • Rotating the lensball drags the visible cap across the source sphere.
   • The seam on a full-sphere lensball is where the cap boundary collapses to a point.
2. (general) — A solid glass sphere held in front of a camera so that the scene appears inside it, refracted, inverted and miniaturised. The photography object the HT sense borrows its name and its picture from: a small ball holding a warped image of a bigger scene, which you turn to look elsewhere.
   • Every travel photographer seems to carry a lensball now.

Etymology

First recorded: 2026-09-14, the Lensball article on the Hei Canon wiki. Coined by: Hei (the technical sense; the photography sense is a 2010s trade name turned generic).

lens

English from the 1690s, from Latin lens (lentil), after the shape of a biconvex piece of glass.

ball

Middle English bal, from Old Norse bǫllr; Germanic root shared with balloon and bowl.

Compound of lens + ball. The photography sense is a 2010s trade name for crystal-ball photography that became generic. The technical sense (2026) keeps the object and adds a state: the ball is a display you rotate, and rotating it is navigation.

Usage

Not a dimensionality-reduction map (global, static, one picture for everything) and not only a magic lens (a movable region laid over a picture): a lensball's own rotation is the navigation, and its transition rule is declared. Say "spherical lensball" or "discrete lensball" when the tier matters.

Sources

• Canonical entry: heidict — Lensball
• Source of truth (YAML): ~/ht/heidict/entries/lensball.yaml
• Curated by Benedict under heidict editorial doctrine.