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Lecture_Chapter4_b

Lecture_Chapter4_b - Review of Thursday Vectors and points...

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Review of Thursday Vectors and points are two/three-dimensional objects Operations on vectors (dot product, cross product, ) Matrices are linear transformations of vectors Objects are transformed by applying transformation matrix to its vertices (coordinate transformation) 29 ECS 175 Chapter 4: Object Transformation a b c d · x y = xa + yb xc + yd
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Review of Thursday Objects are transformed by applying transformation matrix to its vertices (coordinate transformation) 30 ECS 175 Chapter 4: Object Transformation anisotropic scaling 2 0 0 3 · 1 1 = 2 3 2 0 0 3 · 3 3 = 6 9 2 0 0 3 · 4 1 = 8 3 det ( M ) = 6
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A Brief Review of Linear Algebra Rotation 31 ECS 175 Chapter 4: Object Transformation counterclockwise (positive direction of rotation) Determinant: cos α · cos α sin α · ( sin α ) = 1 cos α sin α sin α cos α Inverse M 1 = M T
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A Brief Review of Linear Algebra Constructing a rotation matrix 32 ECS 175 Chapter 4: Object Transformation cos α sin α sin α cos α M (1 , 0) T = (cos α , sin α ) T M (0 , 1) T = ( sin α , cos α ) T ( a, b ) T = a (1 , 0) T + b (0 , 1) T M ( a, b ) T = aM (1 , 0) T + bM (0 , 1) T M ( a, b ) T = ( a cos α b sin α , a sin α + b cos α ) T columns correspond to axes of new coordinate frame (linearity of M)
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A Brief Review of Linear Algebra Shear transformation (shearing) Reflection 33 ECS 175 Chapter 4: Object Transformation 1 0 0 . 5 1 displacement along direction proportional to (signed) distance to a line 1 0 0 1
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A Brief Review of Linear Algebra Applying multiple transformations in sequence 34 ECS 175 Chapter 4: Object Transformation M scal · M rot · v = v ( M scal · M rot ) · v = v Order of operations is important!
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Notes on Matrices in OpenGL 35 ECS 175 Chapter 4: Object Transformation OpenGL <3.0 includes matrix operations and stack Matrix operations and stack deprecated in OpenGL 3.0+ Implement own transformation matrices Pass to vertex shader (OpenGL/GLSL: uniform variables) glMatrixMode( ) glLoadMatrix( ) glMultMatrix( ) glPushMatrix() //push current matrix to top of stack glPopMatrix() //take matrix from top of stack Assignment 2: You may use glLoadMatrix, glMultMatrix or pass the matrices to the vertex shader as uniform variables (do not use glTranslate, glRotate).
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Notes on Matrices in OpenGL 36 ECS 175 Chapter 4: Object Transformation OpenGL <3.0 vertex shader example OpenGL 3.0+ vertex shader example void main(void) { gl_Position = gl_ProjectionMatrix * gl_ModelViewMatrix * gl_Vertex; } attribute vec3 in_Position;
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