Computer Graphics II

Transcription

1 Computer Graphics II Autumn

2 Outline 1 3-D Rotations (contd.)

3 Outline 3-D Rotations (contd.) 1 3-D Rotations (contd.)

4 Amalgamating Steps 2, 3 & 4 Quaternions are a more compact way of solving the task and lend themselves very easily to rotating through an angle θ one or more vectors (and, therefore, points) about an axis u Like previous two methods a translation of the axis to the origin is required first The method of using quaternions is based on Euler-Rodrigues formula for rotation of a vector around an axis (See also, here )

5 Rodrigues Rotation Formula A one-step method that rotates a vector around an arbitrary axis coincident with the origin Reminder: rotating a unit-length vector s = 1 i + 0 j through a positive angle θ gives the vector j s = cos θ i + sin θ j (cos θ, sin θ) i = θ

6 Rotate the vector p around the axis through a positive angle θ project p onto centre axis θ q u = ( p ) it is v = p u that we need to rotate in plane normal to v = p ( p ) p p

7 Rotate the vector p around the axis through a positive angle θ project p onto centre axis θ u = ( p ) it is v = p u that we need to rotate in plane normal to v = p ( p ) p

8 Rotate the vector p around the axis through a positive angle θ project p onto centre axis u = ( p ) it is v = p u that we need to rotate in plane normal to v = p ( p ) u p

9 Rotate the vector p around the axis through a positive angle θ project p onto centre axis u = ( p ) v it is v = p u that we need to rotate in plane normal to v = p ( p ) u p

10 v lies in plane defined by p and The vector s = p is to this plane w lies in this new plane, and so is a scaled multiple of s ; that is, w = k s Though not obvious, k = 1 so w = s = p w = v So v rotated through θ becomes v cos θ + w sin θ p

11 v lies in plane defined by p and The vector s = p is to this plane w lies in this new plane, and so is a scaled multiple of s ; that is, w = k s Though not obvious, k = 1 so w = s = p w = v So v rotated through θ becomes v cos θ + w sin θ p

12 v lies in plane defined by p and The vector s = p is to this plane w lies in this new plane, and so is a scaled multiple of s ; that is, w = k s Though not obvious, k = 1 so w = s = p w = v So v rotated through θ becomes v cos θ + w sin θ w v

13 v lies in plane defined by p and The vector s = p is to this plane w lies in this new plane, and so is a scaled multiple of s ; that is, w = k s Though not obvious, k = 1 so w = s = p w = v So v rotated through θ becomes v cos θ + w sin θ θ

14 Rotation of p through θ around is R(θ,, p ) = u + cos θ v + sin θ w = ( p ) + cos θ( p ( p ) ) + sin θ p = cos θ p + ( p )(1 cos θ) + sin θ p

15 Rotation of p through θ around is R(θ,, p ) = u + cos θ v + sin θ w = ( p ) + cos θ( p ( p ) ) + sin θ p = cos θ p + ( p )(1 cos θ) + sin θ p By noting that v = ( p ) and p = u + v we can rewrite second line above as

16 Rotation of p through θ around is R(θ,, p ) = u + cos θ v + sin θ w = ( p ) + cos θ( p ( p ) ) + sin θ p = cos θ p + ( p )(1 cos θ) + sin θ p By noting that v = ( p ) and p = u + v we can rewrite second line above as R(θ,, p ) = u + cos θ v + sin θ w = ( p v ) + cos θ v + sin θ p = p + (cos θ 1) v + sin θ p = p + (cos θ 1)( ( p )) + sin θ p = p + (1 cos θ) ( p ) + sin θ p (1)

17 Graphical realisation of two previous expressions: First expression Second, more useful, form eqn. (1) previously p q

18 Graphical realisation of two previous expressions: First expression Second, more useful, form eqn. (1) previously

19 Graphical realisation of two previous expressions: First expression Second, more useful, form eqn. (1) previously

20 Euler-Rodrigues Rotation Formula Using trig. identities: sin θ = 2 cos θ 2 sin θ 2 cos θ = 1 2 sin 2 θ 2 1 cos θ = 2 sin2 θ 2 With unit-length (as always) and a = cos θ 2 ω t = sin θ 2 r = (b, c, d) t (not too difficult to show that a 2 + b 2 + c 2 + d 2 = 1) Can now tidy up as follows R(θ,, p ) = p + (1 cos θ)( ( p )) + sin θ( p ) R(a, ω, p ) = p + 2( ω ( ω p )) + 2a( ω p )

21 Euler- We have seen that n = u v can be viewed as the u acting on v, sending it to the vector n. From this point of view u operates on v and in our case we can write s = ω p = M p in terms of matrix multiplication where ( ) 0 d c M = d 0 b, (b, c, d) t = sin θ c b 0 2 Then ω ( ω p ) = M(M p ) = M 2 p R(a, ω, p ) = I p + 2M 2 p + 2aM p = R p

22 Euler- R(a, ω, p ) = R p where a 2 + b 2 c 2 d 2 2(bc ad) 2(bd + ac) R = 2(bc + ad) a 2 + c 2 b 2 d 2 2(cd ab) 2(bd ac) 2(cd + ab) a 2 + d 2 b 2 c 2 Note tr(r) = 3a 2 (b 2 + c 2 + d 2 ) = 4a 2 1 = 2(2a 2 1) + 1 = 2(2 cos 2 θ 2 1) + 1 = 2 cos θ + 1 To verify: det R = 1...

23 Outline 3-D Rotations (contd.) 1 3-D Rotations (contd.)

24 Going the other Way 3-D Rotations (contd.) We ve seen how to construct the rotation matrix Given a matrix, M how do we determine its axis and angle? We can firstly confirm that it is a 3-D rotation matrix by checking that determinant is 1 To find u, the axis, we observe that u is the only vector when rotated that is itself M u = u (M I) u = 0 Solving for this gives u Recall in 3-D, s, the sum of the main diagonal elements (the trace) of the matrix sums to s = cos θ s = M 11 + M 22 + M 33 = cos θ θ = cos 1 (s 1)/2