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• Indicate coordinate systems with every point or matrix – Point: p object – Matrix: M object world • Resulting transformation equation: p camera = (C camera world)‐1 M object world p object • In source code use similar names: – Point: p_object or p_obj or p_o – Matrix: object2world or obj2wld or o2w
We can represent general transformations of homogeneous coordinates by matrices. This idea has been used widely in geometric modeling to describe the relationships between objects.
Lecture L3 - Vectors, Matrices and Coordinate Transformations. By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties.
Make it very explicit what coordinate system is used. Understand how to change coordinate systems. Understand how to transform objects. Understand difference between points, vectors, normals and their coordinates.
The coordinate conversion matrix also provides a quick route to finding the Cartesian components of the three basis vectors of the spherical polar coordinate system. sph
Coordinate Transformations. In this chapter, we explore mappings where a mapping is a function that "maps" one set to another, usually in a way that preserves at least some of the underlyign geometry of the sets. For example, a 2-dimensional coordinate transformation is a mapping of the form. T (u; v) = hx (u; v) ; y (u; v)i.
Define the matrix θ θ θ θ Transformations. • Translation. – P′=T + P • Scale – P′=S ⋅P • Rotation – P′=R ⋅ • We would like all transformations to be multiplications so we can concatenate them • ⇒express points in homogenous coordinates. 12 − = ′ ′ y x y x. sin cos cos sin
