lecture10-1 - Introduction to Computer Graphics CS 445 /...

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Unformatted text preview: Introduction to Computer Graphics CS 445 / 645 Gimbal Lecture 10 Chapter 7: Transformations Overview Rotation representations • Euler • Axis-angle • Quaternion Camera Transformations • Projections Basic 3D Transformations Rotate around Z axis: x' cos Θ − sin Θ y ' sin Θ cos Θ z ' = 0 0 w 0 0 x ' cos Θ y ' 0 Rotate around Y axis: = z ' − sin Θ w 0 Rotate around X axis: x' 1 y ' 0 z ' = 0 w 0 0 1 0 0 0 0 1 0 0 x 0 y 0 z 1w sin Θ 0 x 0 0 y cos Θ 0 z 0 1w 0 0 cos Θ − sin Θ sin Θ cos Θ 0 0 0 x 0 y 0 z 1w 3-D Rotation General rotations in 3-D require rotating about an General arbitrary axis of rotation arbitrary axis Deriving the rotation matrix for such a rotation Deriving directly is a good exercise in linear algebra directly Standard approach: express general rotation as Standard composition of canonical rotations composition canonical • Rotations about X, Y, Z Rotations X, Composing Canonical Rotations Goal: rotate about arbitrary vector A by α Goal: • Idea: we know how to rotate about X,Y,Z Idea: – So, rotate about Y by θ until A lies in the YZ plane So, – Then rotate about X by ϕ until A coincides with +Z Then – Then rotate about Z by α Then – Then reverse the rotation about X (by -ϕ) Then (by – Then reverse the rotation about Y (by -θ) Then (by Composing Canonical Rotations First: rotating about Y by θ until A lies in YZ First: How exactly do we calculate θ ? How • Project A onto XZ plane (Throw away y-coordinate) Project onto • Find angle τ that rotates A to x-axis: Find x-axis θ = -(90° - τ) = τ - 90 ° Second: rotating about x-axis by ϕ until A lies on z-axis Second: z-axis How do we calculate ϕ ? How Composing Matrices So we have the following matrices: • p: The point to be rotated about A by α The • Ryθ : Rotate about Y by θ • Rx ϕ : Rotate about X by ϕ • Rzα : Rotate about Z by α -1 • Rx ϕ -1: Undo rotation about X by ϕ Undo • Ryθ-1 : Undo rotation about Y by θ In what order should we multiply them? Compositing Matrices Short answer: the transformations, in order, are Short written from right to left right left • In other words, the first matrix to affect the vector goes next In to the vector, the second next to the first, etc. to So in our case: 1 p’ = Ryθ --1 Rx ϕ -1 Rzα Rx ϕ Ryθ p p’ Rotation Matrices Notice these two matrices: Rx ϕ : Rotate about X by ϕ -1 Rx ϕ -1: Undo rotation about X by ϕ Undo -1 How can we calculate Rx ϕ -1? Rotation Matrices Notice these two matrices: Rx ϕ : Rotate about X by ϕ -1 Rx ϕ -1: Undo rotation about X by ϕ Undo -1 How can we calculate Rx ϕ -1? • Obvious answer: calculate Rx (-ϕ) • Clever answer: exploit fact that rotation matrices are Clever orthonormal orthonormal Rotation Matrices Notice these two matrices: Rx ϕ : Rotate about X by ϕ -1 Rx ϕ -1: Undo rotation about X by ϕ Undo -1 How can we calculate Rx ϕ -1? • Obvious answer: calculate Rx (-ϕ) • Clever answer: exploit fact that rotation matrices are orthonormal Clever What is an orthonormal matrix? What What property are we talking about? Rotation Matrices Orthonormal matrix: Orthonormal • orthogonal (columns/rows linearly independent) • normalized (columns/rows length of 1) The inverse of an orthogonal matrix is just its The transpose: transpose: −1 T a b c a b c a d h d e f = d e f = b e i h i h i c f j j j Representing 3 Rotational DOFs 3x3 Matrix (9 DOFs) • Rows of matrix define orthogonal axes Euler Angles (3 DOFs) • Rot x + Rot y + Rot z Axis-angle (4 DOFs) • Axis of rotation + Rotation amount Quaternion (4 DOFs) • 4 dimensional complex numbers Rotation Matrices What did we start with? • 3 rotations (roll, pitch, yaw) – Three numbers – Called Euler Angles • Axis-angle – Four numbers And we end up with a 4x4 transformation matrix • Where inner 3x3 describes rotation Really, a whole matrix to represent three numbers? Underconstrained? 9 DOFs must reduce to 3 Rows must be unit length (-3 DOFs) Rows must be orthogonal (-3 DOFs) Drifting matrices is very bad • Numerical errors results when trying to gradually rotate matrix by Numerical adding derivatives adding • Resulting matrix may scale / shear • Gram-Schmidt algorithm will re-orthogonalize your matrix Difficult to interpolate between matrices • Why would we do this? Euler Angles (θ x, θ y, θ z) = RzRyRx • Rotate θx degrees about x-axis Rotate • Rotate θy degrees about y-axis Rotate • Rotate θz degrees about z-axis Rotate Axis order is not defined • (y, z, x), (x, z, y), (z, y, x)… are all legal • Pick one Euler Angles Rotations not uniquely defined • ex: (z, x, y) = (90, 45, 45) = (45, 0, -45) takes positive x-axis to (1, 1, 1) • Cartesian coordinates are independent of one another, but Cartesian Euler angles are not Euler Gimbal Lock • Term derived from mechanical problem that arises in Term gimbal mechanism that supports a compass or a gyro gimbal A Gimbal Hardware implementation of Euler angles (used for Hardware mounting gyroscopes and globes) mounting Gimbal Lock Gimbal Lock Occurs when two axes are Occurs aligned aligned Second and third rotations Second have effect of transforming earlier rotations earlier • ex: Rot x, Rot y, Rot z – If Rot y = 90 degrees, If Rot z == -Rot x Rot Gimbal Lock http://www.anticz.com/eularqua.htm Interpolation Interpolation between two Euler angles is not Interpolation unique unique ex: (x, y, z) rotation • (0, 0, 0) to (180, 0, 0) vs. (0, 0, 0) to (0, 180, 180) • Interpolation about different axes are not Interpolation independent independent Interpolation Axis-angle Notation Define an axis of rotation (x, y, z) and a rotation Define about that axis, θ : R(θ , n) R( 4 degrees of freedom specify 3 rotational degrees degrees of freedom because axis of rotation is constrained to be a unit vector constrained Axis-angle Rotation Given r – Vector in space to rotate n – Unit-length axis in space about which to rotate θ – The amount about n to rotate Solve r’ – The rotated vector nr r’ Axis-angle Rotation Step 1 • Compute rk an extended version of the rotation axis, n • rk = (n ¢ r) n rk r r’ Axis-angle Rotation Compute r? r? = r – (n ¢ r) n r? r r’ Axis-angle Rotation Compute v, a vector perpendicular to rk and r? v = rk £ r? Use v and r? and θ to compute r’ Use v cos(θ) r? + sin(θ) v θ r? Axis-angle Notation rperp = r – (n.r) n θ V = n x (r – (n.r) n) = n x r Rr r rpar = (n.r) n n Rr = Rrpar + Rrperp = Rrpar + (cos θ) rperp + (sin θ) V =(n.r) n + cos θ(r – (n.r)n) + (sin θ) n x r = (cos θ)r + (1 – cos θ) n (n.r) + (sin θ) n x r Axis-angle Notation No easy way to determine how to concatenate No many axis-angle rotations that result in final desired axis-angle rotation desired No simple way to interpolate rotations Quaternion Remember complex numbers: a + ib • Where i2 = -1 Invented by Sir William Hamilton (1843) • Remember Hamiltonian path from Discrete II? Quaternion: • Q = a + bi + cj + dk – Where i2 = j2 = k2 = -1 and ij = k and ji = -k • Represented as: q = (s, v) = s + vxi + vyj + vzk Represented Quaternion A quaternion is a 4-D unit vector q = [x y z w] • It lies on the unit hypersphere x2 + y2 + z2 + w2 = 1 For rotation about (unit) axis v by angle θ For • vector part = (sin θ/2) v = [x y z] vector /2) [x • scalar part = (cos θ/2) scalar /2) =w • (sin(θ/2) vx, sin(θ/2) vy, sin(θ/2) vz, cos (θ/2)) z, cos Only a unit quaternion encodes a rotation - normalize Quaternion Rotation matrix corresponding to a quaternion: • [x y z w] = 1 − 2 y 2 − 2 z 2 [x 2 xy + 2 wz 2 xz − 2wy 2 xy − 2 wz 1 − 2 x 2 − 2 z 2 2 yz + 2 wx 2 xz + 2 wy 2 yz − 2wx 1 − 2 x 2 − 2 y 2 Quaternion Multiplication Quaternion • q1 * q2 = [v1, w1] * [v2, w2] = [(w1v2+w2v1+ (v1 x v2)), w1w2-v1.v2] • quaternion * quaternion = quaternion • this satisfies requirements for mathematical group this group • Rotating object twice according to two different quaternions is equivalent Rotating to one rotation according to product of two quaternions to Quaternion Example X-roll of π X-roll • (cos (π/2), sin (π/2) (1, 0, 0)) = (0, (1, 0, 0)) Y-roll 0f π Y-roll • (0, (0, 1, 0)) Z-roll of π Z-roll • (0, (0, 0, 1)) Ry (π ) followed by Rz (π ) • (0, (0, 1, 0) times (0, (0, 0, 1)) = (0, (0, 1, 0) x (0, 0, 1) = (0, (1, 0, 0)) Quaternion Interpolation Biggest advantage of quaternions • Interpolation • Cannot linearly interpolate between two quaternions Cannot because it would speed up in middle because • Instead, Spherical Linear Interpolation, slerp() • Used by modern video games for third-person Used perspective perspective • Why? SLERP Quaternion is a point on the 4-D unit sphere • interpolating rotations requires a unit quaternion at each step – another point on the 4-D unit sphere • move with constant angular velocity along the great circle between two points move – A great circle is a section of a sphere that contains a diameter of the sphere great (Kern and Bland 1948, p. 87). Any rotation is defined by 2 quaternions, so pick the shortest SLERP Any To interpolate more than two points, solve a non-linear variational To constrained optimization constrained • Ken Shoemake in SIGGRAPH ’85 (www.acm.org/dl) Quaternion Interpolation Quaternion (white) vs. Quaternion Euler (black) interpolation interpolation Left images are linear Left interpolation interpolation Right images are cubic Right interpolation interpolation Quaternion Code http://www.gamasutra.com/features/programming/ 19980703/quaternions_01.htm • Registration required Camera control code • http://www.xmission.com/~nate/smooth.html – File, gltb.c – gltbMatrix and gltbMotion 3D Geometric Primitives Pipeline (for direct illumination) Rendering 3D Modeling Transformation Lighting Transform into 3D world coordinate system Illuminate according to lighting and reflectance Viewing Transformation Transform into 3D camera coordinate system Projection Transformation Transform into 2D screen coordinate system Clipping Scan Conversion Image Clip primitives outside camera’s view Draw pixels (includes texturing, hidden surface, ...) Reminder: Homogeneous Coords What effect does the following matrix have? x ' 1 y ' 0 = z ' 0 w' 0 0 1 0 0 0 x y 0 0 z 0 10w 0 0 1 Conceptually, the fourth coordinate w is a bit like a scale Conceptually, factor factor Homogenous Coordinates Increasing w makes things smaller Increasing We think of homogenous coordinates as defining We a projective space projective • Increasing w “getting further away” Increasing Will come in handy for projection matrices Projection Matrix We talked about geometric transforms, focusing on We modeling transforms modeling • Ex: translation, rotation, scale, gluLookAt() Ex: gluLookAt() • These are encapsulated in the OpenGL modelview matrix These modelview Projection is also represented as a matrix Next few slides: representing orthographic and Next perspective projection with the projection matrix projection Taxonomy of Projections FVFHP Figure 6.10 Taxonomy of Projections Parallel Projection Center of projection is at infinity • Direction of projection (DOP) same for all points View Plane DOP Angel Figure 5.4 Orthographic Projections DOP perpendicular to view plane Front Angel Figure 5.5 Top Side Oblique Projections DOP not perpendicular to view plane DOP not φ = 45 Cavalier o (DOP α = 45 ) φ = 63.4 Cabinet o (DOP α = 63.4 ) H&B Figure 12.24 Orthographic Projection Simple Orthographic Transformation Original world units are preserved • Pixel units are preferred Orthographic: Screen Space Transformation left =10 m right = 20 m top=20 m (max pixx, max pixy) (height in pixels) bottom=10 m (0, 0) (width in pixels) Orthographic: Screen Space Transformation left, right, top, bottom refer to the viewing frustum in left, modeling coordinates modeling width and height are in pixel units This matrix scales and translates to accomplish the This transition in units transition ...
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