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15-462: Computer Graphics

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f(x,y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c 3 4).(,56+1&()+&-$+&78,0(9+,-(6& :).9.-.;+&81+0&.,&<=&9/0+6.,5& :)/5)(91> 3 4).(,56+1&()+&1-/)+0&(1&(& b 1+?8+,*+&/7&-$)++&;+*-/)1@&+(*$& a 0+7.,.,5&(&;+)-+A> 3 B7-+,@&C+&D,/C&.,7/)9(-./,& ('/8-&-$+&;+)-.*+1@&18*$&(1&*/6/)@& -$(-&C+E0&6.D+&-/&.,-+):/6(-+&/;+)& -$+&C$/6+&-).(,56+> !" Barycentric Color Interpolation #$%&'%()'*%(+,)-&(.,',//(0.-%&)12 c 3 4$)'1.567)1&'8%+'&/'0/'&$.1' .-&)(6/7%&./-'.1'!"#$%&'(#)%* %++#,)'"(&-9 3 4$)'-%5)',/5)1':(/5'&$)';())<' 8/(0'!"#.- =$)%>+?'*),%@1)'&$)' b ,//(0.-%&)1'%()'8).A$&1'%11.A-)0' a &/'&$)'>)(&.,)19 3 B/.-&'a /-'&$)'&(.%-A7)'.1'&$)'/(.A.-' /:'&$)'-/-C/(&$/A/-%7',//(0.-%&)' 1+1&)59Goal: Assign a every point (x,y) or (x,y,z) to 3 4$)'>),&/(1':(/5'barycentric coordinatesa &/'b %-0':(/5' (α,β,γa). &/'c %()'&%<)-'%1'*%1.1'>),&/(19 !" Barycentric Coordinates c b a source: http://en.wikipedia.org/wiki/File:Barycentric_coordinates_1.png Solution: x = αa + βb + γc #$%&'%()'*%(+,)-&(.,',//(0.-%&)12 c Some cool properties: 3 4/.-&'p .1'.-1.0)'&$)'&(.%-56)'.7 %-0'/-6+'.7 8'9'! 9'": b 8'9'" 9'": a 8'9'# 9'" 3 ;7'/-)',/<=/-)-&'.1'>)(/:'p .1'/-'%-')05)? 3 ;7'&@/',/<=/-)-&1'%()'>)(/:'p .1'/-'%'A)(&)B? 3 C$)',//(0.-%&)1',%-'*)'D1)0'%1'@).5$&.-5' 7%,&/(1'7/('=(/=)(&.)1'/7'&$)'A)(&.,)1:'6.E)',/6/(? !" Barycentric Color Interpolation c b x a If: x = αa + βb + γc Then: color(x)=αcolor(a)+βcolor(b)+γcolor(c) #$%&'()*%+','--%.+)$*(/ !"#$%&'#()*+,#-.$+/0 1 !'23+(/4'2*5('-*67*!#(8+/4#2 1 !'23+(/4'2*5('-*97*!#(8+/4#2 !" Transformations Translation,Translation, rotation,rotation, scalingscaling 2D2D 3D3D HomogeneousHomogeneous coordinatescoordinates TransformingTransforming normalsnormals ExamplesExamples Shirley Chapter 6 Uses of Transformations • Modeling – build complex models by positioning simple components – transform from object coordinates to world coordinates • Viewing – placing the virtual camera in the world – specifying transformation from world coordinates to camera coordinates • Animation – vary transformations over time to create motion CAMERA OBJECT WORLD Computer Graphics 15-462 3 Rigid Body Transformations Rotation angle and line about which to rotate Computer Graphics 15-462 4 Non-rigid Body Transformations Distance between points on object do not remain constant Computer Graphics 15-462 5 Basic 2D Transformations Scale Shear chalkboard Rotate Computer Graphics 15-462 6 Composition of Transformations • Created by stringing basic ones together, e.g. – “translate p to the origin, rotate, then translate back” can also be described as a rotation about p • Any sequence of linear transformations can be collapsed into a single matrix formed by multiplying the individual matrices together • Order matters! • Can apply a whole sequence of transformations at once chalkboard 0 1 2 3 Translate to the origin, rotate, then translate back. Computer Graphics 15-462 7 3D Transformations • 3-D transformations are very similar to the 2-D case • Scale • Shear • Rotation is a bit more complicated in 3-D – different rotation axes chalkboard Computer Graphics 15-462 8 Fixed Euler Angles for 3-D Rotations • Independent rotations about each coordinate axis – angle interpolation for animation generates bizarre motions – rotations are order-dependent, and there are no conventions about the order to use • Widely used anyway, because they're “simple” Computer Graphics 15-462 9 Euler Angles for 3-D Rotations Computer Graphics 15-462 10 Other representations of 3D orientation • Relative rather than fixed Euler angles • Axis/Angle: rotate by ! about axis V • Quaternions: – a generalization of complex numbers –3 give the rotation axis - magnitude is sin !/2 –1 gives cos !/2 (the amount of rotation) –unit magnitude - points on a 4-D unit sphere Computer Graphics 15-462 11 But what about translation? •Translation is not linear--how to represent as a matrix? chalkboard Computer Graphics 15-462 12 But what about translation? •Translation is not linear--how to represent as a matrix? chalkboard •Trick: add extra coordinate to each vector •This extra coordinate is the homogeneous coordinate, or w •When extra coordinate is used, vector is said to be represented in homogeneous coordinates •We call these matrices Homogeneous Transformations Computer Graphics 15-462 13 W? Where did that come from? • Practical answer: –W is a clever algebraic trick. –Don’t worry about it too much. The w value will be 1.0 for the time being (until we get to perspective viewing transformations) • More complete answer: –(x,y,w) coordinates form a 3D projective space. –All nonzero scalar multiples of (x,y,1) form an equivalence class of points that project to the same 2D Cartesian point (x,y). –For 3-D graphics, the 4D projective space point (x,y,z,w) maps to the 3D point (x,y,z) in the same way. Computer Graphics 15-462 14 Homogeneous 2D Transformations The basic 2D transformations become Translate: Scale: Rotate: !" %" !" %" 10tx sx 00 !cos" ( )sin( 0%" $" '" $" '" 01ty 0 sy 0 $"sin( cos( 0'" $" '" $" '" $" 001'" #"001&" #"001&" #" &" Now any sequence of translate/scale/rotate operations can be combined into a single homogeneous matrix by multiplication. 3D transforms are modified similarly Computer Graphics 15-462 15 Rigid Body Transformations Rotation angle and line about which to rotate Computer Graphics 15-462 16 Rigid Body Transformations •A transformation matrix of the form xx xy tx yx yy ty 001 where the upper 2x2 submatrix is a rotation matrix and column 3 is a translation vector, is a rigid body transformation. •Any series of rotations and translations results in a rotation and translation of this form (and no change in the distance between vertices) Computer Graphics 15-462 17 Sequences of Transformations x' x' x' x' x' x' x' x' x' TRANSFORMED POINTS M PARAMETERS Often the same MATRICES transformations are applied M to many points Calculation time for the M matrices and combination is negligible compared to that of transforming the points x x x x x x x x x x UNTRANSFORMED Reduce the sequence to a POINTS single matrix, then transform Computer Graphics 15-462 18 Collapsing a Chain of Matrices. • Consider the composite function ABCD, i.e. p’ = ABCDp • Matrix multiplication isn’t commutative - the order is important • But matrix multiplication is associative, so can calculate from right to left or left to right: ABCD = (((AB) C) D) = (A (B (CD))). • Iteratively replace either the leading or the trailing pair by its product M ! D M ! A M ! CM or M ! MB both give the same result. M ! BM M ! MC M ! AM M ! MD Premultiply Postmultiply Computer Graphics 15-462 19 What is a Normal?– refresher Indication of outward facing direction for lighting and shading Order of definition of vertices in OpenGL Right hand rule Computer Graphics 15-462 20 Transforming Normals • It’s tempting to think of normal vectors as being like porcupine quills, so they would transform like points • Alas, it’s not so. • We need a different rule to transform normals. chalkboard Computer Graphics 15-462 21 Examples • Modeling with primitive shapes Computer Graphics 15-462 22 Announcements • Reading for Thursday: Shirley Ch: 2,6, & 7 • Project 1 due next Tuesday... • Style: 10% • Any questions? • Bulletin Board Mini-Demonstration.
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  • A Tutorial on Euler Angles and Quaternions

    A Tutorial on Euler Angles and Quaternions

    A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ Version 2.0.1 c 2014–17 by Moti Ben-Ari. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Chapter 1 Introduction You sitting in an airplane at night, watching a movie displayed on the screen attached to the seat in front of you. The airplane gently banks to the left. You may feel the slight acceleration, but you won’t see any change in the position of the movie screen. Both you and the screen are in the same frame of reference, so unless you stand up or make another move, the position and orientation of the screen relative to your position and orientation won’t change. The same is not true with respect to your position and orientation relative to the frame of reference of the earth. The airplane is moving forward at a very high speed and the bank changes your orientation with respect to the earth. The transformation of coordinates (position and orientation) from one frame of reference is a fundamental operation in several areas: flight control of aircraft and rockets, move- ment of manipulators in robotics, and computer graphics. This tutorial introduces the mathematics of rotations using two formalisms: (1) Euler angles are the angles of rotation of a three-dimensional coordinate frame.