2DTransform

# 2DTransform - 2D G Geometric ti Transformations T f ti...

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2D Geometric Transformations Chapter 5 tro to Computer Graphics Intro. to Computer Graphics Spring 2009, Y. G. Shin

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troduction Introduction ± We deal a lots with objects defined in 2-D and 3-D worlds in computer graphics ll objects have shape position and orientation ± All objects have shape, position and orientation ± Geometry is the study of the relationships among bjects objects ± Modeling + rendering : computer programs that describe these objects and how light bounces around to illuminate them in order to calculate the final pixel values on the display. wo types of eometry operations ± Two types of geometry operations ± Vertex operations - Operate on individual vertexes ± Primitive operations - Operate on all the vertexes of a p p primitive
Coordinate-Free Geometry FG style of expressing geometric objects and ± CFG - A style of expressing geometric objects and relations that do not rely on any specific ordinate system coordinate system ± Representing geometry in terms of coordinates can frequently lead to physical confusion

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Example of coordinate-dependence Point p Point q ± What is the “sum” of these two positions ?
If you assume coordinates, … p = (x 1 , y 1 ) q = (x 2 , y 2 ) ± The sum is (x 1 +x 2 , y 1 +y 2 ) ± Is it correct ? ± Is it geometrically meaningful ?

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If you assume coordinates, … (x 1 +x 2 , y 1 +y 2 ) p = (x 1 , y 1 ) q = (x 2 , y 2 ) Origin ± Vector sum ± (x 1 , y 1 ) and (x 2 , y 2 ) are considered as vectors from the origin to p and q , respectively.
If you select a different origin, … (x 1 +x 2 , y 1 +y 2 ) p = (x 1 , y 1 ) q = (x 2 , y 2 ) 9 The geometric relationship of the result of adding rigin two points depends on the coordinate system. There is no clear geometric interpretation for Origin ± If you choose a different coordinate frame, you will adding two points. get a different result

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Points and Vectors Point q vector (q-p) Point p ± A scalar is just a real number. a location in space It does not have any ± A point is a location in space. It does not have any intrinsic coordinates. a direction and a magnitude. It may be ± A vector is a direction and a magnitude. It may be specified as the difference between two points.
FG Operations CFG Operations ± magnitude of a vector : oint ector addition: v r r ± point-vector addition: 1 2 1 2 1 1 p p v p v p = = + r ± vector addition : 3 2 1 v v v r r r = + ± vector scaling : ot product : 1 v r α s r r r r ± dot product : θ cos 2 1 2 1 v v v v =

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CFG Operations ± cross product: θ sin 2 1 2 1 v v v v r r r r = × ± Linear combination of vectors : v v r r = α ± Affine combination of points : i i i 1 if , = = i i i i i p p ± 0 if , = = i i i i i v p r
The Cross Product The length is also equal to the area of the parallelogram whose sides are given by and v r v r r r r 1 2 2 1 v v × 2 v r r r 2 1 area v v × = 1 v

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Various Geometry Operations - Scaling coordinate means multiplying each of ± Scalinga coordinate means multiplying each of its components by a scalar × 2 X × 2, Y × 0.5
Scaling ± Scaling operation: y ax x = ' ' ± r, in matrix form: by y = Or, in matrix form: x a x 0 '

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2DTransform - 2D G Geometric ti Transformations T f ti...

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