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PolySlerp1DxS3

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PolySlerp1DxS3 is the official sample program of the paper "PolySlerp: Spherical Polygon Linear Interpolation" [1] — an interactive FireMonkey (Delphi) application for constructing and comparing curves of unit quaternions — smooth one-parameter families of 3D rotations — on the unit 3-sphere $S^3$. Five spline schemes (Linear, Bézier, B-Spline, Catmull-Rom, Lanczos) are combined with six spherical weighted-averaging ("barycenter") operators (GLerp, Slerp, PowSlerp, PolySlerp, ExpMap, ModExpMap), and two independently configured curves are rendered simultaneously over a textured globe for direct visual comparison.

利用ライブラリ

• LUX ：The LUXOPHIA standard library providing basic mathematical types such as vectors, matrices, and quaternions.
• LUX.FMX.Graphics.D3 ：A 3D graphics helper library built on the FireMonkey framework.
• LUX.Sphere ：A spherical geometry library for curves and barycenter operations on the spheres $S^2$ and $S^3$.

1. Overview

• Control loops: closed loops of $N \in {4, 6, 8, 12, 20}$ control quaternions (TPolyPoins3S04 … TPolyPoins3S20). Each loop is derived from a closed loop of $N$ points on the 2-sphere (TPolyPoins2S04 … TPolyPoins2S20): every anchor starts as the rotation carrying the pole vector $\hat{y}$ onto its loop point, composed with a random twist about $\hat{y}$; the twists are then relaxed by iterated degree-4 B-spline smoothing (built from repeated slerp steps, TPolyPoins3S.BSpline4), re-pinning each axis to its loop point on every pass, until the loop of orientations converges.
• Curve = spline scheme × barycenter operator: every spline scheme is written so that each evaluation is a weighted barycenter of control quaternions. The averaging rule is pluggable — the same Bézier or B-spline weights can be averaged by six different spherical methods, which produce visibly different curves.
• Two curves at once: an Upper Curve and a Lower Curve are configured independently. Their sample dots are drawn as complementary hemisphere caps (THemiPoinsUp3D / THemiPoinsLo3D), so the two curves remain distinguishable even where they coincide.
• Rotation visualization: a quaternion $q$ is displayed at the point $R,(q,\hat{y},q^{-1})$ on a globe of radius $R$, as a small textured sphere oriented by the full rotation of $q$ — the position shows the image of the rotation axis and the texture orientation shows the twist around it.
• Arc-length-uniform sampling: TPlots3S resamples each curve at equal angular steps, with the step (PlotGap) chosen from the dot size so that successive dots just touch.
• Twist stress-test: a slider applies an alternating twist of $\pm\tau$ ($\tau \in [-90^\circ, +90^\circ]$) to consecutive control quaternions about their local axes, exposing the different behaviors of the averaging operators.

2. Mathematical Background

2.1 Unit quaternions and rotations

A unit quaternion $q = w + x,\mathbf{i} + y,\mathbf{j} + z,\mathbf{k}$ with $\lVert q \rVert = 1$ lies on the 3-sphere $S^3 \subset \mathbb{H}$ and acts on a vector $v \in \mathbb{R}^3$ (a pure imaginary quaternion) as a rotation [5]:

R_q(v) = q\,v\,q^{-1} \tag{1}

The application identifies the type TDouble3S (a point of $S^3$) with the quaternion type TDoubleQ, and displays $q$ at $R \cdot R_q(\hat{y})$ on the globe (TDoubleQ.Trans, Scene.pas).

2.2 Slerp and weighted slerp

Spherical linear interpolation between $p, q \in S^3$ with $\cos\theta = \langle p, q \rangle$ is [2][4]

\mathrm{Slerp}(p, q; t) = \frac{\sin\bigl((1-t)\,\theta\bigr)\,p + \sin(t\,\theta)\,q}{\sin\theta} \tag{2}

The code (LUX.S3.Curve.Slerp) generalizes this to weighted points $(p, w_1)$, $(q, w_2)$ with arbitrary real weights. With $W = w_1 + w_2$:

(p, w_1) \oplus (q, w_2)
= \left( \frac{\sin\bigl(\tfrac{w_1}{W}\theta\bigr)\,p + \sin\bigl(\tfrac{w_2}{W}\theta\bigr)\,q}{\sin\theta},\; W \right) \tag{3}

which equals $\mathrm{Slerp}(p, q;, w_2/W)$ carrying the combined weight $W$. Nearly parallel inputs fall back to normalized linear interpolation. Signed weights are supported (TDouble3Sw negates the weight under unary minus), which is required by the Catmull-Rom and Lanczos schemes whose basis functions have negative lobes.

2.3 Exponential and logarithm maps

Identifying pure imaginary quaternions with the Lie algebra $\mathfrak{so}(3)$, the exponential map $\exp : \mathfrak{so}(3) \to S^3$ and its inverse are

\exp(\theta\,\hat{n}) = \cos\theta + \hat{n}\,\sin\theta,
\qquad
\ln\bigl(\cos\theta + \hat{n}\,\sin\theta\bigr) = \theta\,\hat{n} \tag{4}

LUX.Quaternion implements the general versions on $\mathbb{H}$ ($\ln q = \ln\lVert q \rVert + \operatorname{atan2}(\lVert \mathbf{v} \rVert, w),\hat{\mathbf{v}}$, and the corresponding $\exp$, with Pow(q, t) = Exp(t·Ln(q))).

2.4 Barycenter operators

Each curve evaluation reduces to a weighted barycenter $\mathcal{B}\bigl({(q_i, w_i)}_{i=1}^{n}\bigr)$ of unit quaternions. Six operators are implemented (LIB.Bary.S3.* delegating to LUX.S3.Curve.*):

\mathcal{B}_{\mathrm{GLerp}} = \frac{\sum_i w_i\,q_i}{\bigl\lVert \sum_i w_i\,q_i \bigr\rVert} \tag{5}

(p, w_1) \oplus_{\mathrm{pow}} (q, w_2) = \Bigl( p\,\bigl(p^{-1} q\bigr)^{w_2 / (w_1+w_2)},\; w_1 + w_2 \Bigr),
\qquad q^{t} = \exp\bigl(t \ln q\bigr) \tag{6}

PolySlerp (PolySlerp1D) removes the order dependence of the chained fold by a binary-pyramid recursion — the type-2 (interior-correction) summation of [1], i.e. PolySlerp-2: at each level the interior weights are halved and adjacent pairs are combined with $\oplus$ from (3),

P^{(L)}_i = \tilde{P}^{(L+1)}_i \oplus \tilde{P}^{(L+1)}_{i+1},
\qquad
\tilde{P}^{(L+1)}_i = \begin{cases} P^{(L+1)}_i & i \in \{0, L+1\} \\ \tfrac{1}{2} P^{(L+1)}_i & \text{otherwise} \end{cases} \tag{7}

as illustrated by the weight-flow diagram in the source (Fig. 3(c) of [1]):

Weight flow for n = 4 points A???D ??? each node is the ??? of its two children,
"/k" is the factor the node is scaled by before it is fed to its parent
(end entries /1, interior entries /2, per the tilde in eq. (7))

???4A+4B+4C+4D       ????????? level 0 : result
  ??????4A+3B+1C /1   ????????? level 1
  ???  ??????4A+2B /1  ????????? level 2
  ???  ???  ??????4A /1 ????????? level 3 : control point A
  ???  ???  ??????4B /2 ????????? level 3 : control point B
  ???  ??????2B+2C /2  ????????? level 2
  ???     ??????4B /2
  ???     ??????4C /2
  ??????1B+3C+4D /1   ????????? level 1
     ??????2B+2C /2   ????????? level 2 : the same node as above (shared)
     ???  ??????4B /2
     ???  ??????4C /2
     ??????2C+4D /1   ????????? level 2
        ??????4C /2   ????????? level 3 : control point C
        ??????4D /1   ????????? level 3 : control point D

ExpMap averages in the log domain (cf. the exp/log curve constructions of [3]); as noted in [1], the result depends on the base point of the projection, so it is only an approximate average:

\mathcal{B}_{\mathrm{ExpMap}} = \exp\!\left( \frac{1}{W} \sum_i w_i \ln q_i \right),
\qquad W = \sum_i w_i \tag{8}

ModExpMap first computes the GLerp mean $g$ from (5) and averages in the tangent space at $g$ — exactly the base-point choice recommended in [1] — largely removing the bias of (8):

\mathcal{B}_{\mathrm{ModExpMap}} = g\,\exp\!\left( \frac{1}{W} \sum_i w_i \ln\bigl(g^{-1} q_i\bigr) \right) \tag{9}

2.5 Spline schemes

All schemes (LIB.Curve.*, generic over the point type) express one segment as a barycenter of consecutive control points. Where two algorithms are offered, Basis Function evaluates one $n$-ary barycenter with basis weights, while Recursive Lerp applies the classical pairwise recursion using binary barycenters — for non-flat averaging operators these are not equivalent, and the difference can be inspected in the viewer.

C(t) = \mathcal{B}\Bigl( \bigl\{ \bigl(p_{i+j},\; b_{j,n}(t)\bigr) \bigr\}_{j=0}^{n} \Bigr) \tag{10}

2.6 Arc-length resampling

TPlots3S measures the length of a curve as the accumulated angle between consecutive rotated axis vectors, $d(q_0, q_1) = \angle\bigl(R_{q_0}(\hat{y}),, R_{q_1}(\hat{y})\bigr)$, over a 1024-edge table, then places sample points at equal arc-length intervals PlotGap $= 2 \arcsin(r_{\mathrm{dot}} / R)$, so that adjacent dots exactly touch on screen.

3. Architecture

[1] Ownership ??? Main.pas

???TForm1
  ??????_Poins3S :TPolyPoins3S     ????????? control-quaternion loop
  ??????_Bary3S :TDoubleBary3S ??6 ????????? the six barycenter operators
  ??????_Curve3S0 :TCurve3S ??5    ????????? the five spline schemes (Lower Curve)
  ??????_Curve3S1 :TCurve3S ??5    ????????? the five spline schemes (Upper Curve)
  ??????_Plots3S0 :TPlots3S        ????????? arc-length resampling of _Curve3S0
  ??????_Plots3S1 :TPlots3S        ????????? arc-length resampling of _Curve3S1

[2] Scene graph ??? Scene.pas, inside TViewport3D (LUX.Sphere.FMX / LUX.S3.FMX)

???TWorld3D
  ??????TCamera3D
  ???  ??????TLight3D
  ??????TSphere3D                  ????????? globe
  ??????TPoins3D                   ????????? ??? _Poins3S
  ??????THemiPoinsLo3D             ????????? ??? _Plots3S0
  ??????THemiPoinsUp3D             ????????? ??? _Plots3S1

[3] Data flow ??? Main.pas model ??? Scene.pas mesh

???_Poins3S :TPolyPoins3S
  ??????TPoins3D                   ????????? control-point dots
  ??????_Curve3S0 :TCurve3S ??5
  ???  ??????_Plots3S0 :TPlots3S
  ???     ??????THemiPoinsLo3D      ????????? lower-hemisphere dots
  ??????_Curve3S1 :TCurve3S ??5
     ??????_Plots3S1 :TPlots3S
        ??????THemiPoinsUp3D       ????????? upper-hemisphere dots

[4] Class hierarchy ??? control points, sampling (Q = TDouble3S, unit quaternion)

???TPoins<Q>
  ??????TLoopPoins<Q>
  ???  ??????TPolyPoins3S
  ???     ??????TPolyPoins3S04      ????????? (4 points)
  ???     ??????TPolyPoins3S06      ????????? (6 points)
  ???     ??????TPolyPoins3S08      ????????? (8 points)
  ???     ??????TPolyPoins3S12      ????????? (12 points)
  ???     ??????TPolyPoins3S20      ????????? (20 points)
  ??????TPlots<Q>
     ??????TPlots3S                ????????? arc-length resampler

[5] Class hierarchy ??? curves and barycenter operators

???TCurve<Q>
  ??????TCurveAlgo<Q>              ????????? Bary property references a TBarycenter<Q>
     ??????TCurveLinear<Q>         ????????? (TCurveLinear3S)
     ??????TCurveBezier<Q>         ????????? (TCurveBezier3S)
     ??????TCurveBSpline<Q>        ????????? (TCurveBSpline3S)
     ??????TCurveCatmullRom<Q>     ????????? (TCurveCatmullRom3S)
     ??????TCurveLanczos<Q>        ????????? (TCurveLanczos3S)

???TBarycenter<Q> (TDoubleBary3S)
  ??????TBaryGLerp3S
  ??????TBarySlerp3S
  ??????TBaryPowSlerp3S
  ??????TBaryPolySlerp3S
  ??????TBaryExpMap3S
  ??????TBaryModExpMap3S

???PolySlerp1DxS3/
  ??????PolySlerp1DxS3.dpr         ????????? program entry
  ??????PolySlerp1DxS3.dproj       ????????? RAD Studio project (Win32 / Win64)
  ??????Main.pas / Main.fmx        ????????? main form: UI controls and model wiring
  ??????Scene.pas                  ????????? 3D scene construction (globe, dots, camera)
  ??????_DATA/                     ????????? textures (globe, dot markers; Git LFS)
  ??????_LIBRARY/                  ????????? vendored libraries (reference only)
     ??????Bary/                   ????????? TBarycenter<> and the six S?? operators
     ??????Curve/                  ????????? generic spline schemes and S?? aliases
     ??????LUXOPHIA/
        ??????LUX/                 ????????? scalar/vector/quaternion/basis functions
        ??????LUX.Sphere/          ????????? S??/S?? point grids, curves, FMX meshes
        ??????LUX.FMX.Graphics.D3/ ????????? 3D mesh shaper base classes

4. Usage / Controls

The Upper Curve is drawn with upper-hemisphere dots, the Lower Curve with lower-hemisphere dots; the larger dots are the shared control points.

5. Building

1. Requirements: RAD Studio (Delphi) with the FireMonkey framework; the project file uses format version 20.4. Target platforms defined in PolySlerp1DxS3.dproj: Win32 and Win64.
2. Clone with Git LFS enabled (git lfs install) — the textures in _DATA/ are stored in LFS.
3. Open PolySlerp1DxS3.dproj, select a platform, and run. All library units are vendored under _LIBRARY/; no external packages are required.
4. The application loads textures from ..\..\_DATA\ relative to the executable, which matches the default output directory layout (e.g. Win64\Debug\). Keep that relative location if you move the binary.

6. References

1. M. Nakayama and I. Fujishiro, "PolySlerp: Spherical Polygon Linear Interpolation", The Journal of the Institute of Image Electronics Engineers of Japan, 54(4), pp. 384–393, 2025 (in Japanese). [PDF]
2. K. Shoemake, "Animating Rotation with Quaternion Curves", SIGGRAPH '85, pp. 245–254, 1985.
3. M.-J. Kim, M.-S. Kim, S. Y. Shin, "A General Construction Scheme for Unit Quaternion Curves with Simple High Order Derivatives", SIGGRAPH '95, pp. 369–376, 1995.
4. Wikipedia: Slerp
5. Wikipedia: Quaternions and spatial rotation

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