Line–Cyclide Intersection and Colinear Point Quadruples in the Double Conformal Model

Abstract

In this paper, we look at using the double conformal model for ray tracing and for tool positioning in Computer Numerically Controlled (CNC) machining. In particular, we explore the intersection of a line with a cyclide in the double conformal model, and how to extract the four points from the resulting colinear point quadruple. Further, we show how to directly construct a colinear point quadruple from four points, and we show how to find the line containing the points of a colinear point quadruple. We also briefly touch on barycentric coordinates and affine combinations in DCGA. Finally, we discuss applications of the double conformal model in ray tracing and in CNC machining.

Supported by NSERC.

Appendices

A Derivation of Affine Combination

An affine combination of two points yields a sphere; the reflection of the point at infinity yields the center of this sphere [2]. In this appendix, we generalize the idea of the affine combination of two CGA points to the affine of an arbitrary number of CGA points.

Lemma A1

Given n vectors ${{\textbf{p}}_{c1}},\dots ,{{\textbf{p}}_{cn}}$, n CGA points, ${\boldsymbol{P}_{C1}},\dots ,{\boldsymbol{P}_{Cn}}$ with ${\boldsymbol{P}_{Ci}}={{\mathcal {D}}1}({{\textbf{p}}_{ci}})$ and n scalars $a_1,\dots ,a_n$ where $\sum _{i=1}^n a_i=1$. Then the affine combination

$$\begin{aligned} \boldsymbol{s} = \sum _{i=1}^n a_i {\boldsymbol{P}_{Ci}} \end{aligned}$$

is a sphere centered at $\sum _{i=1}^n a_i{{\textbf{p}}_{ci}}$ with radius

$$\begin{aligned} \sum _{0<i<j\le n} a_ia_j |{{\textbf{p}}_{ci}}-{{\textbf{p}}_{cj}}|^2. \end{aligned}$$

Proof:

$$\begin{aligned} \boldsymbol{s} = & {} \sum a_i {\boldsymbol{P}_{Ci}}\\ = & {} \sum a_i(\boldsymbol{e}_o + {{\textbf{p}}_{ci}} + {\frac{1}{2}}|{{\textbf{p}}_{ci}}|^2 \boldsymbol{e}_{\infty })\\ = & {} \boldsymbol{e}_o + \sum a_i{{\textbf{p}}_{ci}} + \sum {\frac{1}{2}}a_i |{{\textbf{p}}_{ci}}|^2 \boldsymbol{e}_{\infty }\\ = & {} \boldsymbol{e}_o + \sum a_i{{\textbf{p}}_{ci}} + {\frac{1}{2}}\left| \sum a_i{{\textbf{p}}_{ci}}\right| ^2 \boldsymbol{e}_{\infty } \\ {} & {} - {\frac{1}{2}}\left[ \left| \sum a_i{{\textbf{p}}_{ci}}\right| ^2 - \sum a_i|{{\textbf{p}}_{ci}}|^2 \right] \boldsymbol{e}_{\infty }. \end{aligned}$$

We see that $\boldsymbol{s}$ has the form of a sphere centered at $\sum a_i{{\textbf{p}}_{ci}}$. Expanding the coefficient of $-{\frac{1}{2}}\boldsymbol{e}_{\infty }$ of the last term (the radius of $\boldsymbol{s}$ squared) in coordinates gives

$$\begin{aligned} \left| \sum a_i{{\textbf{p}}_{ci}}\right| ^2 - \sum a_i|{{\textbf{p}}_{ci}}|^2 = & {} \sum a_i^{\,2}{{\textbf{p}}_{ci}}^{\,2} + \sum _{i<j} 2a_ia_j ({{\textbf{p}}_{ci}}\cdot {{\textbf{p}}_{cj}}) - \sum a_i {{\textbf{p}}_{ci}}^{\,2}\\ = & {} \sum a_i(a_i-1){{\textbf{p}}_{ci}}^{\,2}+ \sum _{i<j} 2a_ia_j ({{\textbf{p}}_{ci}}\cdot {{\textbf{p}}_{cj}})\\ = & {} -\sum a_i\Big (\sum _{j\ne i}a_j\Big ){{\textbf{p}}_{ci}}^{\,2}+ \sum _{i<j} 2a_ia_j ({{\textbf{p}}_{ci}}\cdot {{\textbf{p}}_{cj}})\\ = & {} -\sum _{i,j\ne i} a_ia_j{{\textbf{p}}_{ci}}^{\,2}+ \sum _{i<j} 2a_ia_j ({{\textbf{p}}_{ci}}\cdot {{\textbf{p}}_{cj}})\\ = & {} \sum _{i,j> i} \left( -a_ia_j {{\textbf{p}}_{ci}}^{\,2} +2a_ia_j ({{\textbf{p}}_{ci}}\cdot {{\textbf{p}}_{cj}})-a_ia_j{{\textbf{p}}_{cj}}^{\,2}\right) \\ = & {} -\sum _{i<j} a_ia_j ({{\textbf{p}}_{ci}}-{{\textbf{p}}_{cj}})^2. \end{aligned}$$

Note that $\boldsymbol{e}_o$ and $\boldsymbol{e}_{\infty }$ in this section are 1-blades.

B Affine Combination in DCGA

We can compute an affine combination of DCGA points by converting them to CGA1 and CGA2 points, computing an affine combination of these CGA points (given a CGA1 sphere and a CGA2 sphere), and taking the outer product of these two sphere.

Theorem B1

Given n DCGA points, ${\boldsymbol{P}_{D1}},\dots ,{\boldsymbol{P}_{Dn}}$ with ${\boldsymbol{P}_{Di}=\boldsymbol{P}_{C^1_i}\wedge \boldsymbol{P}_{C^2_i}}$ and n scalars $a_1,\dots ,a_n$ where $\sum _{i=1}^n a_i=1$. Then the affine combination

$$\begin{aligned} \boldsymbol{s} = \boldsymbol{s}_1\wedge \boldsymbol{s}_2 = & {} \sum _{i=1}^n a_i {\boldsymbol{P}_{C^1_i}}\wedge \sum _{i=1}^n a_i {\boldsymbol{P}_{C^2_i}} \\ = & {} \sum _{i=1}^n a_i (-{\boldsymbol{P}_{Di}}\cdot \boldsymbol{e}_{\infty 2})\wedge \sum _{i=1}^n a_i (-{\boldsymbol{P}_{Di}}\cdot \boldsymbol{e}_{\infty 1}) \\ = & {} a_1^2{\boldsymbol{P}_{D1}}+a_2^2{\boldsymbol{P}_{D2}}+\dots a_n^2{\boldsymbol{P}_{Dn}}+a_1a_2{\boldsymbol{P}_{C^1_1}}{\boldsymbol{P}_{C^2_2}}+\dots +a_ia_j{\boldsymbol{P}_{C^1_i}}{\boldsymbol{P}_{C^2_j}} \end{aligned}$$

is a DCGA sphere centered at $\sum _{i=1}^n a_i{{\textbf{p}}_{c^1_i}}\wedge \sum _{i=1}^n a_i{{\textbf{p}}_{c^2_i}}$ with radius

$$\begin{aligned} \sum _{0<i<j\le n} a_ia_j |{{\textbf{p}}_{c^1_i}}-{{\textbf{p}}_{c^1_j}}|^2. \end{aligned}$$

$$\begin{aligned} \boldsymbol{s}_1 = & {} \sum a_i {\boldsymbol{P}_{C^1_i}}\\ = & {} \sum a_i(\boldsymbol{e}_{o1} + {{\textbf{p}}_{c^1_i}} + {\frac{1}{2}}|{{\textbf{p}}_{c^1_i}}|^2 \boldsymbol{e}_{\infty 1})\\ = & {} \boldsymbol{e}_{o1} + \sum a_i{{\textbf{p}}_{c^1_i}} + \sum {\frac{1}{2}}a_i |{{\textbf{p}}_{c^1_i}}|^2 \boldsymbol{e}_{\infty 1}\\ = & {} \boldsymbol{e}_{o1} + \sum a_i{{\textbf{p}}_{c^1_i}} + {\frac{1}{2}}\left| \sum a_i{{\textbf{p}}_{c^1_i}}\right| ^2 \boldsymbol{e}_{\infty 1} \\ {} & {} - {\frac{1}{2}}\left[ \left| \sum a_i{{\textbf{p}}_{c^1_i}}\right| ^2 - \sum a_i|{{\textbf{p}}_{c^1_i}}|^2 \right] \boldsymbol{e}_{\infty 1}\\ \boldsymbol{s}_2 = & {} \sum a_i {\boldsymbol{P}_{C^2_i}}\\ = & {} \sum a_i(\boldsymbol{e}_{o2} + {{\textbf{p}}_{c^2_i}} + {\frac{1}{2}}|{{\textbf{p}}_{c^2_i}}|^2\boldsymbol{e}_{\infty 2})\\ = & {} \boldsymbol{e}_{o2} + \sum a_i{{\textbf{p}}_{c^2_i}} + \sum {\frac{1}{2}}a_i |{{\textbf{p}}_{c^2_i}}|^2\boldsymbol{e}_{\infty 2}\\ = & {} \boldsymbol{e}_{o2} + \sum a_i{{\textbf{p}}_{c^2_i}} + {\frac{1}{2}}\left| \sum a_i{{\textbf{p}}_{c^2_i}}\right| ^2\boldsymbol{e}_{\infty 2} \\ {} & {} - {\frac{1}{2}}\left[ \left| \sum a_i{{\textbf{p}}_{c^2_i}}\right| ^2 - \sum a_i|{{\textbf{p}}_{c^2_i}}|^2 \right] \boldsymbol{e}_{\infty 2}\\ \boldsymbol{s}_1\wedge \boldsymbol{s}_2 = & {} {\boldsymbol{P}'_{C1}}\wedge {\boldsymbol{P}'_{C2}} -{\frac{1}{2}}{\boldsymbol{P}'_{C1}}R^2\boldsymbol{e}_{\infty 2}-{\frac{1}{2}}{\boldsymbol{P}'_{C2}}R^2 \boldsymbol{e}_{\infty 1}+\frac{1}{4}R^4 \boldsymbol{e}_{\infty } \end{aligned}$$

where

$$\begin{aligned} {\boldsymbol{P}'_{C1}} = & {} \boldsymbol{e}_{o1} + \sum a_i{{\textbf{p}}_{c^1_i}} + {\frac{1}{2}}\left| \sum a_i{{\textbf{p}}_{c^1_i}}\right| ^2 \boldsymbol{e}_{\infty 1}\\ {\boldsymbol{P}'_{C2}} = & {} \boldsymbol{e}_{o2} + \sum a_i{{\textbf{p}}_{c^2_i}} + {\frac{1}{2}}\left| \sum a_i{{\textbf{p}}_{c^2_i}}\right| ^2\boldsymbol{e}_{\infty 2}\\ R^2 = & {} \left| \sum a_i{{\textbf{p}}_{c^1_i}}\right| ^2 - \sum a_i|{{\textbf{p}}_{c^1_i}}|^2 \end{aligned}$$

As in CGA, we can reflect the DCGA point at infinity through this DCGA sphere to obtain the point form of the affine combination: $\boldsymbol{s}\,\boldsymbol{e}_{\infty }\,\boldsymbol{s}^{-1}$.