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CLASS 4 CS770/870

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Translation

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Scale

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Multiplying Matrices. The R C rule What happens when we do two translates? What happens when we do two scales? What happens when we translate and scale, or scale then translate (Commutative?) Exercise

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Matrices in GL virtual machine OpenGL maintains a matrix stack glPushMatrix() creates pushes a new 4x4 matrix on the top of the stack containing a copy of the existing top of stack matrix. glTranslate, glScale and glRotate cause a new matrix to be contructed and post multiplied by the top of the stack.

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The matrix stack glLoadIdentity();M1 I glRotatef();M1 IR1 glTranslatef();M1 IR1T1 glScalef();M1 IR1T1S1 glPushMatrix();M2 M1 glTranslatef();M2 M1T2 glPushMatrix();M3 M2 M1 M2

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The view window to viewport transformation In open GL As a set of matrixes

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Rotation about z axis

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Derive rotation about z Take notes Rotations about x and y.

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Exercise: do it by drawing Given a square glRectf(-1,-1,1,1) glRotatef(-30.0,0,0,1); glScalef(2.0,1.0,1.0); gLTranslatef(2.0,0.0,0.0); glRectf (-1,-1,1,1); glTranslatef(0.0,2.0,0.0); glScalef(1.0,2.0,1.0); glRotatef(45,0.0,0.0, 1.0); glRectf (-1,-1,1,1);

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The Scene window to viewport mapping Scene window box (Lsw, Rsw, Bsw, Tsw) Viewport box (Lvp, Rvp, Bvp, Tvp); Exercise specify open gl commands to accomplish this) If time construct the matrix (Just the top row)

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Some basic linear algebra (CH 4) Dot and Cross Product. A dot product of unit vectors gives the cosine of the angle between two unit vectors a. b = (a 1 *b 1 + a 2 *b 2 + a 3 *b 3 ) = |a||b|cos( |a| = Sqrt(a 1 *a 1 + a 2 *a 2 + a 3 *a 3 ); a/|a| = a 1 /|a| + a 2/ |a| + a 3 /|a| // unit vector

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Cross Product of two vectors a x b = (a 1, a 2, a 3 ) T x (b 1, b 2, b 3 ) The result is a vector = (a 2 * b 3 ) – (a 3 * b 2 ), (a 3 * b 1 ) – (a 1 * b 3 ), (a 1 * b 2 ) – (a 2 * b 1 ) If both vectors are unit vectors the result is at right angles to the plane running through the other two.

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2IV60 Computer Graphics 2D transformations

2IV60 Computer Graphics 2D transformations

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