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A method of calculating intersection of quadratic surfaces in quaternion algebra

Przemys law Dobrowolski

Warsaw University of Technology Faculty of Mathematics and Information Science

Be ¸dlewo, 24-26 May 2013

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Introduction

An exact motion planning problem can be formulated in terms of computational geometry. In case of 3-dimensional motion planning with rotations there exists the following problem:

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Introduction

An exact motion planning problem can be formulated in terms of computational geometry. In case of 3-dimensional motion planning with rotations there exists the following problem: Problem Given two non-translated quadratic surfaces in algebra of unit quaternions, compute an exact parametrization of the intersection

• f the two surfaces.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Introduction

An exact motion planning problem can be formulated in terms of computational geometry. In case of 3-dimensional motion planning with rotations there exists the following problem: Problem Given two non-translated quadratic surfaces in algebra of unit quaternions, compute an exact parametrization of the intersection

• f the two surfaces.

Non-translated quadratic surface in algebra of unit quaternions will be defined later.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Introduction

General features: each non-translated quadratic surface is defined by 10 coefficients corresponding configuration space is homeomorphic to S3 with antipodal points identified ambient space is 4-dimensional

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Quaternion algebra

Quaternion algebra H An algebra of hyper-complex numbers of form: q = ai + bj + ck + d (1) which base elements i, j, k have the following multiplication rules: i2 = j2 = k2 = ijk = −1 is called quaternion algebra.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Quaternion algebra

Quaternion algebra H An algebra of hyper-complex numbers of form: q = ai + bj + ck + d (1) which base elements i, j, k have the following multiplication rules: i2 = j2 = k2 = ijk = −1 is called quaternion algebra. Note: Multiplication in quaternion algebra is non-commutative!

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Quaternion algebra

We are going to use unit quaternions which can be used to represent a rotation in 3-space:

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Quaternion algebra

We are going to use unit quaternions which can be used to represent a rotation in 3-space: Subalgebra of unit quaternions The subalgebra U of H such that for each q ∈ U: q∗q = 1 where q∗ = −ai − bj − ck + d is called subalgebra of unit quaternions. Topology: Subalgebra of unit quaternions is a Lie algebra. It is a two-fold universal cover of SO(3) which is topologically homeomorphic to S3 with antipodal points defined.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Quadratic surface in quaternion algebra

A non-translated quadratic surface in H can be written as a quadratic form in R4. Assuming that q = ai + bj + ck + d ∈ H and its corresponding vector ¯ q = [a, b, c, d]T ∈ R4 there is a function ¯ S : R4 − → R: ¯ S(¯ q) = ¯ qTM¯ q = 0 where M =

    

a11 a12 a13 a14 a12 a22 a23 a24 a13 a23 a33 a34 a14 a24 a34 a44

    

which is equivalent to function S : H − → R: S(q) =a11a2 + a22b2 + a33c2 + a44d2+ 2(a12ab + a13ac + a14ad + a23bc + a24bd + a34cd) = 0 it will be called shortly H-quadric.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Intersection of quadratic surfaces in quaternion algebra

The following theorem allows a person to use a simpler Cartesian intersection routines instead of hyper-complex one. Theorem Let Γ(ξ) = [Γx(ξ), Γy(ξ), Γz(ξ), Γw(ξ)]T be the intersection of two quadratic surfaces in homogeneous space P3 for ξ ∈ P1. Intersection of two H-quadrics is equal to: ¯ q(ξ) = ± Γ(ξ) Γ(ξ) (2)

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

Proof Assume that in H there are given two H-quadrics S1 and S2 with matrices M1 and M2. The intersection is a set of quaternions ¯ q = [a, b, c, d]T satisfying:

    

¯ qT ¯ q = 1 ¯ qTM1¯ q = 0 ¯ qTM2¯ q = 0 It is impossible that all of a, b, c, d are equal to zero simultaneously because of the first equation. Assume for now that d is non-zero. The second and the third equation can be divided by d2, resulting in:

• tTM1t = 0

tTM2t = 0 (3)

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

The newly introduced vector t is equal to [ a

d , b d , c d , 1]T. It can be

• bserved that the two equations of (3) form a quadric intersection

problem in R3. Important note: both tTM1t and tTM2t are not necessarily quadratic forms. Since t4 is equal to 1, each formula may contain the parameter t in the first order as well as scalars. As a result, a person must consider an intersection of two general quadrics in R3. This problem can be effectively solved. Both quadrics are given in terms of tx = a

d , ty = b d , tz = c d . Now, we

assume that the intersection curve Γ(ξ) = [Γx(ξ), Γy(ξ), Γz(ξ), Γw(ξ)]T is in homogeneous coordinates: tx = Γx(ξ) Γw(ξ), ty = Γy(ξ) Γw(ξ), tz = Γz(ξ) Γw(ξ), ξ ∈ P1

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

To recover all of a, b, c, d, first d is computed. Summing up the squares of tx, ty and tz one obtains: t2

x + t2 y + t2 z = a2

d2 + b2 d2 + c2 d2 = a2 + b2 + c2 d2 = 1 − d2 d2 = 1 d2 − 1 so, t2

x + t2 y + t2 z + 1 = 1 d2 and d2 = 1 t2

x +t2 y +t2 z +1

by plugging in the intersection curve Γ, one can write: d2 = 1

Γx(ξ)2 Γw(ξ)2 + Γy(ξ)2 Γw(ξ)2 + Γz(ξ)2 Γw(ξ)2 + Γw(ξ)2 Γw(ξ)2

= Γw(ξ)2 Γx(ξ)2 + Γy(ξ)2 + Γz(ξ)2 + Γw(ξ)2 A rotation by a quaternion q is identified with a rotation by a quaternion −q. Hence, in the above equation a square root can be taken of both sides without a loss of generality.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

Finally one obtains: d = Γw(ξ) Γ(ξ) where Γ(ξ) =

• Γx(ξ)2 + Γy(ξ)2 + Γz(ξ)2 + Γw(ξ)2. The

remaining quaternion coordinates are: a = dtx = Γw(ξ) Γ(ξ) Γx(ξ) Γw(ξ) = Γx(ξ) Γ(ξ) b = dty = Γw(ξ) Γ(ξ) Γy(ξ) Γw(ξ) = Γy(ξ) Γ(ξ) c = dtz = Γw(ξ) Γ(ξ) Γz(ξ) Γw(ξ) = Γz(ξ) Γ(ξ)

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

The above formulas can be finally rewritten as the H-quadric intersection parametrization: q(ξ) = ±Γx(ξ)i + Γy(ξ)j + Γz(ξ)k + Γw(ξ) Γ(ξ)

• r, equivalently in a vector form:

¯ q(ξ) = ±[Γx(ξ), Γy(ξ), Γz(ξ), Γw(ξ)]T Γ(ξ) ¯ q(ξ) = ± Γ(ξ) Γ(ξ) where Γ := intersect(M1, M2).

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Proof

A note is also needed about initial choice of d as the coordinate by which the remaining coordinates were divided. Because it is not possible that all of a, b, c, d are simultaneously zero, it is possible to non-constructively divide the H space into four subspaces in which the selected quaternion component is non-zero. In each of these fragments, the proof is repeated, regards to different quaternion component.

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Conclusions

intersection of non-translated quadrics on S3 sphere is not more difficult than intersection of general quadrics in P3 (nevertheless, it is still a complex problem!) an ingredient for other motion planning algorithms (involving 3D rotations)

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra

Thank you

Przemys law Dobrowolski Intersection of quadratic surfaces in quaternion algebra