Lecture_Chapter4_b

Ecs 175 chapter 4 object transformation 51 review of

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Unformatted text preview: rld 0. 5 0 = 0 0 0 0.5 0 0 0 0 0.5 0 p1 p2 · vlocal p3 1 Local coordinate frame (source) expressed in global coordinates (target). x x y and vectors v = y Transforms points p = z z 1 0 ECS 175 Chapter 4: Object Transformation 50 Review of Tuesday •  Origin of current coordinate system is fixed point Identical transformation matrix has different effects based on relative position of the object and coordinate origin. ECS 175 Chapter 4: Object Transformation 51 Review of Tuesday •  More complex operations can be performed by concatenating transformations. For example: vworld = T (p)Rz (α)T (−p) · vlocal Rotation around arbitrary point (2D) A = T ( p) R x ( − β 1 ) R y ( − β 2 ) R z ( α ) R y ( β 2 ) R x ( β 1 ) T ( − p) Rotation around arbitrary axis (3D) ECS 175 Chapter 4: Object Transformation 52 Sequences of Transformations •  How do we interpret a sequence of transformation matrices? •  We can read transformations from right to left or from left to right •  Reading from right to left: Interpret operations as being performed in (fixed) world coordinate system. •  Reading from left to right: Interpret operations as being performed in (dynamic) object coordinate system. ECS 175 Chapter 4: Object Transformation 53 Object Transformation •  The vertex processor transforms vertices by applying transformation of current transformation matrix setTransformationMatrix(matrix1) renderObject(object1) //matrix1 is active setTransformationMatrix(matrix2) renderObject(object2) //matrix2 is active Pseudo code Caution when using OpenGL matrix operations (OpenGL <3.0): OpenGL does matrix post-multiplication as opposed to pre-multiplication. ECS 175 Chapter 4: Object Transformation 54 Coordinate Transformations - Summary •  Homogeneous coordinates allow us to •  Incorporate local coordinate system origins (translation) •  Distinguish points and vectors •  Do a full transform between coordinate systems •  Important notions •  Points are different from vectors (cf. vertices and normals) •  Order of transformations matters •  Rotation and translation are rigid-body transformations •  Programming: be aware of row-major vs. column major matrices ECS 175 Chapter 4: Object Transformation 55 Chapter 4 - Summary •  Object geometry (vertices and connectivity) is passed to the graphics pipeline together with transformation operations •  Transformation operations are represented by matrices •  Objects are transformed by applying the transformation to each of its vertices (vertex processor does this in parallel) •  The transformed objects need to be projected into a 2D coordinate system and mapped to the screen (“Where do objects end up on screen?” - next chapter) ECS 175 Chapter 4: Object Transformation 56...
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