Four USD quaternion

Recently, in rewriting the scene management module of the game engine, I have been reviewing some knowledge about the four-dollar number and doing some simple notes here.

Four can be used to accurately describe the rotation of three-dimensional vector, and can effectively express the superposition of multiple rotation operations, so in the three-dimensional game engine scene management module, the four-dollar number has a very important significance.

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first, the definition

form a = ai + bJ + Ck + D is a complex number called four, where i,J,k is an imaginary number (a primitive that is called a four-tuple), A, B, C, and D are real numbers.

second, the common nature

1. I2 = J2 = k2 =-1

2. ij = k JK = i ki = j

3. ij =–ji jk =–kj ki =-ki

4. II* = 1 I* =–i i.e. i* and i conjugate,J,K the same

5. Multiplication of four-tuple satisfies the binding law and distributive law, and does not satisfy the commutative law

6. The imaginary part of four yuan is regarded as three-dimensional vector, then the vector part of two four is αβ =-α,β +αxβ, so that four yuan a = α + d1,b = C7>β + D2, then

AB =-α•beta +αxbeta + d2α + D1Beta + d1d2 = (d1a2–c1b2 + B1C2 + a1d2) i
+ (c1a2 + d1b2–a1c2 + b1d2) J
+ (-b1a2 + a1b2 + d1c2 + c1d2) k
–a1a2–b1b2–c1c2 + d1d2

7. (AB) * = b*a*

8. Define the norm of four yuan A = ai + bJ + Ck + D for: | | a| | = A2 + b2 + C2 + d2 , modulo: | a| = sqrt (A2 + b2 + C2 + D2)

9. Define the inverse of the four-dollar a as: A-1 = */| | a| |

A-m = (A-1) m = (Am)-1

three, the use of four yuan to express the vector rotation

Assuming that the vector α revolves around the hinge e = (xe,ye,ze) Rotates the θ angle to get β, then:

β = uαu-1

which

u = e sin (θ/2) + cos (Θ/2)

u-1 = u* =- e sin (θ/2) + cos (Θ/2)

Therefore, we can use the four-tuple u = (x,y,z,w) to represent coordinate rotations, where:

x = sin (Θ/2) XE

y = sin (θ/2) ye

z = sin (θ/2) ze

w = cos (Θ/2)

iv. using matrices to represent coordinate rotations

Assuming that the axis of revolution is a = (Xa,ya,za) and the rotation angle is α, the rotation matrix is as follows:

five or four-dollar number and transformation of rotation matrix

According to the half-angle formula:

sinα= 2sin (Α/2) cos (Α/2)

cosα= Cos2 (Α/2)-sin2 (Α/2)

Cos2 (Α/2) = (1 +cosα)/2

Sin2 (Α/2) = (1-cosα)/2

A four-dollar conversion to a rotation matrix can be expressed as follows:

Four USD quaternion