7 Rotate About Arbitrary Point Say you wish to rotate about the point (a,b)You know how to rotate about (0,0)Translate so that (a,b) is at (0,0)x’=x–a, y’=y–bRotatex”=(x-a)cos-(y-b)sin, y”=(x-a)sin+(y-b)cosTranslate back againxf=x”+a, yf=y”+b
8 Scaling an Object not at the Origin What happens if you apply the scaling transformation to an object not at the origin?Based on the rotating about a point composition, what should you do to resize an object about its own center?
9 Back to Rotation About a Pt Say R is the rotation matrix to apply, and p is the point about which to rotateTranslation to Origin:Rotation:Translate back:The translation component of the composite transformation involves the rotation matrix. What a mess!
10 Homogeneous Coordinates Use three numbers to represent a point(x,y)=(wx,wy,w) for any constant w0Typically, (x,y) becomes (x,y,1)To go backwards, divide by wTranslation can now be done with matrix multiplication!
12 Homogeneous Transform Advantages Unified view of transformation as matrix multiplicationEasier in hardware and softwareTo compose transformations, simply multiply matricesOrder matters: AB is generally not the same as BAAllows for non-affine transformations:Perspective projections!Bends, tapers, many others
13 Directions vs. Points We have been talking about transforming points Directions are also important in graphicsViewing directionsNormal vectorsRay directionsDirections are represented by vectors, like points, and can be transformed, but not like points(1,1)(-2,-1)
14 Transforming Directions Say I define a direction as the difference of two points: d=a–bThis represents the direction of the line between two pointsNow I translate the points by the same amount: a’=a+t, b’=b+tHow should I transform d?
15 Homogeneous Directions Translation does not affect directions!Homogeneous coordinates give us a very clean way of handling thisThe direction (x,y) becomes the homogeneous direction (x,y,0)The correct thing happens for rotation and scaling alsoScaling changes the length of the vector, but not the directionNormal vectors are slightly different – we’ll see more later
16 3D Transformations Homogeneous coordinates: (x,y,z)=(wx,wy,wz,w) Transformations are now represented as 4x4 matricesTypical graphics packages allow for specification of translation, rotation, scaling and arbitrary matricesOpenGL: glTranslate[fd], glRotate[fd], glScale[fd], glMultMatrix[fd]
18 3D RotationRotation in 3D is about an axis in 3D space passing through the originUsing a matrix representation, any matrix with an orthonormal top-left 3x3 sub-matrix is a rotationRows are mutually orthogonal (0 dot product)Determinant is 1Implies columns are also orthogonal, and that the transpose is equal to the inverse
20 Problems with Rotation Matrices Specifying a rotation really only requires 3 numbersAxis is a unit vector, so requires 2 numbersAngle to rotate is third numberRotation matrix has a large amount of redundancyOrthonormal constraints reduce degrees of freedom back down to 3Keeping a matrix orthonormal is difficult when transformations are combinedRotations are a very complex subject, and a detailed discussion is way beyond the scope of this course
21 Alternative Representations Specify the axis and the angle (OpenGL method)Hard to compose multiple rotationsEuler angles: Specify how much to rotate about X, then how much about Y, then how much about ZHard to think about, and hard to composeAny three axes will do e.g. X,Y,XSpecify the axis, scaled by the angleOnly 3 numbers, called the exponential mapQuaternions
22 Quaternions 4-vector related to axis and angle, unit magnitude Rotation about axis (nx,ny,nz) by angle :Reasonably easy to composeReasonably easy to go to/from rotation matrixOnly normalized quaternions represent rotations, but you can normalize them just like vectors, so it isn’t a problemEasy to perform spherical interpolation
23 Other Rotation Issues Rotation is about an axis at the origin For rotation about an arbitrary axis, use the same trick as in 2D: Translate the axis to the origin, rotate, and translate back againRotation is not commutativeRotation order mattersExperiment to convince yourself of this
24 Transformation Leftovers Scale, shear etc extend naturally from 2D to 3DRotation and Translation are the rigid-body transformations:Do not change lengths or angles, so a body does not deform when transformed
25 Modeling 101For the moment assume that all geometry consists of points, lines and facesLine: A segment between two endpointsFace: A planar area bounded by line segmentsAny face can be triangulated (broken into triangles)
26 Modeling and OpenGLIn OpenGL, all geometry is specified by stating which type of object and then giving the vertices that define itglBegin(…) …glEnd()glVertex[fdv]Three or four components (regular or homogeneous)Float, double or vector (eg float)Chapter 2 of the red book
27 RenderingGenerate an image showing the contents of some region of spaceThe region is called the view volume, and it is defined by the userDetermine where each object should go in the imageViewing, ProjectionDetermine which pixels should be filledRasterizationDetermine which object is in front at each pixelHidden surface elimination, Hidden surface removal, VisibilityDetermine what color it isLighting, Shading
28 Graphics PipelineGraphics hardware employs a sequence of coordinate systemsThe location of the geometry is expressed in each coordinate system in turn, and modified along the wayThe movement of geometry through these spaces is considered a pipelineLocal Coordinate SpaceWorld Coordinate SpaceView SpaceCanonical View VolumeDisplay Space