SUMMARY. CS380: Introduction to Computer Graphics Track-/Arc-ball Chapter 8. Min H. Kim KAIST School of Computing 18/04/06.

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1 8/4/6 CS38: Introduction to Computer Graphics Track-/Arc-ball Chapter 8 Min H. Kim KAIST School of Computing Quaternion SUMMARY 2

2 8/4/6 Unit norm quats. == rotations Squared norm is sum of 4 squares. Any quaternion of the form cos θ 2 cos θ sin θ 2 2 x sin θ = sin θ 2 2 y sin θ 2 z has a unit form Conversely, any such unit norm quaternion can be interpreted (along with its negation as a unique rotation matrix. ω ĉ = ω 2 + x 2 + y 2 + z 2, ĉ =[x, y,z] t ˆk = ω ĉ 3 Operations Quat * quat multiply ω ĉ ω 2 ĉ 2 = (ω ω 2 ĉ ĉ 2 (ω ĉ 2 + ω 2 ĉ + ĉ ĉ 2 Where and are the dot and cross product on 3 dimensional coordinate vectors. Correctly models (rotation matrix * (rotation matrix multiplications Trackball example: M = Matrix4::makeXRotation(-dy * Matrix4::makeYRotation(dx; 4 2

3 8/4/6 3 Unit norm quats Identity rotation example Flip rotation example Unit quaternion multiplication 5 ˆ, ˆ ˆk, ˆk ĉ ĉ 2 = ĉ ĉ 2 ĉ ĉ 2 ˆk ˆk 2 ˆk ˆk 2 = ˆk 2 ˆk Operations scalar * quat multiply NB Quaternions q and q are the same rotation!!! Unit quaternion multiplicative inverse (conjugate 6 cos θ 2 sin θ 2 = cos θ 2 sin θ 2, ˆ = cos θ 2 sin θ 2 cos θ 2 sin θ 2 α ω ĉ = αω αĉ, example - ω ĉ = ω ĉ

4 8/4/6 Rotate a vector by a unit quaternion Perform the following triple quaternion multiplication: cos θ cos θ q 2 = v' 2 cˆʹ sin θ ĉ sin θ 2 v 2 v'= qvq We need inv(q additionally, such that q and v are not orthogonal! If q and v are orthogonal, v =qv. ĉ ĉ 2 (ω ĉ 2 +ω 2 ĉ +ĉ ĉ 2 If we just multiply q with v, qv cannot become a vector (the first element will be non-zero. The additional right multiplication of inverse allows us to get a rotated vector. But we will write this in code as: cvec = quat * cvec ω ω 2 = (ω ω 2 ĉ ĉ 2 7 LERP (Linear Interpolation An even easier hack is to do 4D Linear interpolation (LERP and renormalization a α p p = a +α(b a p = a +α! v (-α p = ( αa +αb! v b cos θ cos θ ( α sin θ + α sin θ 8 4

5 8/4/6 Power-based SLERP Spherical Linear Interpolation α (-α a p α p! v p = a +α(b a! v b q p =! vq = rq q! v = rq cos θ! v = sin θ α v! = α θ = αθ 2 r p = v! α q = ( rq α q cos αθ! 2 v α = sin αθ 2 p = a +α! v p = ( αa +αb 9 Power-based SLERP cn(: ω ĉ In order to select the short interpolation of less than 8 degrees, When we interpolate, before calling the power operator, we first check the sign of the first coordinate, and conditionally negate the quaternion. We call this the conditional negation operator Quaternions q and q are the same rotation!!! Finally, we output: cos θ cos θ cn sin θ sin θ α cos θ sin θ = ω ĉ [ ω, ĉ] [ω,ĉ] 5

6 8/4/6 Sphere-based SLERPing In any dimension n, a trigonometric argument can be used to show that spherical linear interpolation between any two unit vectors in! n, can be calculated as: sin[( α Ω] sin(ω sin[( αω]! v sin(ω + sin[αω]! sin(ω where Ω = cos (! v! v cos θ sin θ + sin[αω] sin(ω v cos θ sin θ (RBT * RBT and inv(rbt The result is a new rigid transform with translation t + r t 2 and rotation r r 2 Use this to code up the * operator. Mind the Cvec3s (the t s and Cvec4s (needed for q*v. i t r i t 2 r 2 = 3D vector i t + r t 2 4D vector r r 2 i t r = i r t r 2 6

7 8/4/6 More code In GLSL, you will still use its matrix data type. The only Matrix4s (that will survive are the projmatrix, the MVM and the NMVM, which get sent to your shaders. Also, when we need to do object scaling, we cannot capture this in an RigTform, so this will also be an Matrix4 used in creating the MVM. To communicate with the vertex shader using 4- by-4 matrices, we need a procedure maketranslation(rigtform and Matrix4 quattomatrix (RigTform to use T * R, which turns quaternions into a 4-by-4 rotation matrix. 3 Chapter 8 BALLS: TRACK AND ARC 4 7

8 8/4/6 Track and arc balls How should we link mouse motion to object rotation Can do better than our current setup Want the feeling of pushing a sphere around (trackball Want path invariance (arcball Reminders: Affine transform: A affine = TL Rigid body transform: A RBT = TR 5 Track and arc balls 6 8

9 8/4/6 Setup We are moving an object with respect to cubeeye a t = w t (O T (E R The user clicks on the screen and drags the mouse. We wish to interpret this user motion as some rotation M that is applied to o t with respect to a t 7 Mental model Imagine a sphere of some chosen radius that is centered at o, the origin of o t. User clicks on the screen at screen pixel s over the sphere in the image We interpret this as the user selecting some 3D point p on the sphere. The user then moves the mouse to other screen pixel over the s sphere, (!p We interpret as a second point p 2 on the sphere 8 9

10 8/4/6 Mental model Define the unit direction vectors v, v 2 : normalize ( p o and normalize ( p 2 o respectively. Define the angle φ = arccos( v v 2 Define the axis k = normalize( v v 2 s (!p 9 Mental model Let s try to formulate the rotation about the axis k can be represented by the quaternion. Where ˆk, ˆv, ˆv 2 are the coordinates 3-vectors representing the vectors k, v, v 2 with respect to the frame a t. s (!p 2

11 8/4/6 Ball rotation using quaternions What we want Derivation Therefore! v 2 = q! v, q =! v 2! v such that q and! v are orthogonal, and q and! v 2 are orthogonal. ˆv 2 ˆv = ˆv ˆv 2 ˆv ˆv = cos(φ 2 sin(φˆk cos θ 2 cos(φ q = sin θ = sin(φˆk. 2 s (!p φ = θ 2, θ =2φ. 2 Ball rotation using quaternions Rotation of 2φ degrees about the axis k can be represented by the quaternion. cos(φ sin(φˆk = ˆv ˆv 2 ˆv ˆv 2 ˆk, ˆv, ˆv 2 Where are the coordinates 3-vectors representing the vectors with respect to the frame. = ˆv 2 ˆv k, v, v 2 a t s (!p 22

12 8/4/6 Ball rotation using quaternions Arcball: M is the rotation of 2φ degrees about the axis k. Trackball: is the rotation of φ degrees about M the axis k. Could be implemented with matrices or quaternions. Arcball is very easy with quaternions s (!p 23 Properties Trackball feels like the user is simply grabbing a physical point on a sphere and dragging it around. But s to, followed by to s 3 is different from moving directly from s to s 3 p will be rotated to p 3, but the results can differ by some twist about the axis o p 3. This path dependence also exists in our simple rotation interface. s (!p Arcball: the object appears to spin twice faster than expected. s 3 (!p

13 8/4/6 Arcball is Path independent If we compose two arcball rotations, corresponding to motion from p to p 2 followed by motion from p to p 2, we get 3 ˆv 2 ˆv 3 ˆv ˆv 2 ˆv 2 ˆv 3 ˆv ˆv 2 Read right to left, global in the unchanging a t frame ˆv 3 ˆv 2 ˆv 2 ˆv = ˆv 3 ˆv = ˆv ˆv 3 ˆv ˆv 3 which is exactly what we would have gotten had we moved directly from to. p p 3 25 Implementation Trackball and Arcball can be directly implemented using either 4-by-4 matrices or quaternions to represent the transformation M. We will use quaternions, since we already have them. The resulting quaternion depends only on vector ˆv So origin of frame is irrelevant We can work in eye coordinates instead of cubeeye. 26 3

14 8/4/6 Getting eye coordinates One slightly tricky part is computing the coordinates of the point on the sphere corresponding to a selected pixel This is geometric ray tracing (this is essentially raytracing, which we will cover later Hack: work in window coordinates X-axis is the horizontal axis of the screen, the y-axis is the vertical axis of the screen, and the z-axis is coming out of the screen. s (!p Think of the sphere s center as simply sitting on the screen. 27 Getting eye coordinates Given the (x,y window coordinates of click, the z coordinate on the sphere can be solved using (x c x 2 + (y c y 2 + (z 2 = r 2 z = r 2 (x c x 2 (y c y 2 [c x,c y,] t are the window coordinates of the center of the sphere r is the radius of the sphere measured in pixels if outside of the sphere, then clamp to its boundary All we need is normalized ˆv, so just normalized such vectors s (!p 28 4

15 8/4/6 Calculation Need the center of the sphere So we give you code that transforms eye coordinates to screen coordinates Cvec2 getscreenspacecoord(const Cvec3& p, const Matrix4& projection, double frustnear, double frustfovy, int screenwidth, int screenheight We draw the ball using object coordinates, so we need to calculate its size in eye/object coordinates So we provide you with double getscreentoeyescale(double z, double frustfovy, int screenheight In the ball drawer, you right multiply a scale matrix to the MVM 29 Translation NB the distance after perspective projection should be different from the distance in the object coordinates. In translation, we interpret mouse displacement (measured in pixels to object displacement. May as well use the same screentoeyescale factor so the object moves with the mouse Once the object is moved, or we change the eye we need to recalculate the scale Wait for click up. 3 5