Quaternionic Bézier curves, surfaces and volume

. We extended the rational Bézier construction for linear, bi-linear and three-linear map, by allowing quaternion weights. These objects are Möbius invariant and have halved degree with respect to the real parametrization. In general, these parametrizations are in four dimensional space. We analyse when a special the three-linear parametrized volume is in usual three dimensional subspace and gives three orthogonal family of Dupine cyclides.


Introduction
A linear rational Bézier curve with quaternionic control points and weights is a circle (see [3]). A bi-linear rational Bézier surface with quaternion control points and weights is a cyclide (see [2]). In this note we define and analyse a special tri-linear quaternionic rational map. This could be useful in geometric modelling and discreet differential geometry (see [1]).

Notations and definitions
We denote by R, C, H the set of real numbers, complex numbers and quaternion numbers respectively.
In general, the quaternion set H can be represented as We denote real and imaginary parts of quaternion q = [r, p] by Re(q) = r, Im(q) = p. The multiplication in the algebra H is defined as where p 1 · p 2 , p 1 × p 2 are scalar and vector products in R 3 . We denote byq = [r, −p] a conjugate quaternion to q = [r, p], |q| = r 2 + p · p = √ qq is the length of the quaternion, q −1 =q/|q| 2 = [r/|q| 2 , −p/|q| 2 ] denote the multiplicative inverse of q, i.e. qq −1 = q −1 q = 1. Denote the set of pure imaginary quaternions

Quaternion rational linear Bézier curve
We represent a curve in Bézier form with quaternionic control points a k ∈ Im H and weights w k ∈ H, k = 0, 1. Formally speaking, we are dealing with a quaternion function in homogeneous coordinates (a k w k , w k ) ∈ H 2 , k = 0, 1. The quaternion rational linear Bézier curve is defined in as customary quotient: Here we consider n(t), d(t) as quaternions, d(t) −1 is an inverse quaternion and n(t)d(t) −1 is the multiplication of two quaternions. So, in general, we have c h (t) ∈ H = R 4 .

Remark 1.
If we change the weights w 0 , w 1 to 1, w 1 w −1 0 the parametrized curve c h (t) is the same. Moreover, if we change the parameter t to s = ρt/(1 − t + ρt) (ρ ∈ R) and weights w 0 , w 1 to w 0 , w 1 /ρ then the curve is the same too.
A quaternionic rational Bézier curve c(t) of degree one is a circular arc with two endpoints a 0 , a 1 . This case is well understood (see [3]). We note that these curves are invariant with respect to Möbius transformation.

Bi-linear quaternion Bézier surfaces
In this section we consider a special bi-linear quaternionic Bézier surface. We remind how to construct a principal Dupin cyclide patch. For detailed description we recommended the preprint [4].
Let us consider four points Q = {a 0 , a 1 , a 3 , a 2 } cyclically arranged on a circle and an unit orthogonal frame of two vectors {v 1 , v 2 } at the point a 0 . We will use notations: Using the triple data T = {Q, v 1 , v 2 } we compute the following weights Let denote by H(T ) = {(a i , w i ), i = 0, 1, 2, 3} a collection of points and weights. We define a special bi-linear quaternion surface One can prove that such that the surface patch D H(T ) (s, t) is in R 3 (see [4]). Moreover, this patch is on some Dupine cyclide. We notice that the weights computed by formulas (9) are not unique. If we change weights (w 0 , w 1 , w 2 , w 3 ) with weights (w 0 q, λw 1 q, µw 2 q, λµw 3 q), q ∈ H, λ, µ ∈ R we get the same patch with different parametrization (see [4]).
Similarly, we see that .
Since we have the following identities for angles Multiplication of right sides gives We can apply a distance formula for inversion: |A ′ B ′ | = r 2 |AB|/(|OA||OB|), where O is a center and r is a radius of inversion; A ′ , B ′ are points obtained after inversion of the points A, B. From the equality (17) using the distance formula we get This is the first identity in Lemma 1. Analogous, we have identities similar to the identity (17) and using the distance formula we get identities (13), (14). Now we explain how to prove that the weight w 123 defined by the formula (11) is equal to w 123 defined by the another formula (15). The proof of the formula (16) is similar. First of all using the property (E) for the normalized difference (see [ If we compare two real coefficient in the formula (11) and (15) we see that they are equal because of the identity (13).
For the simplicity of notation in tri-linear map we will use the following indexing for points and weights