Презентация Geometric Transformations онлайн

Содержание слайда: Geometric Transformations Spring, 2018 AUA

Содержание слайда: Intro & General Information

Содержание слайда: General Information Transformation of a point is basic in GT. It can be formulated as follows: Given a point P that belongs to a geometric model find the corresponding point P* in the new position such that P* = f(P, transformation parameters) The transformation parameters should provide ONE-TO-ONE-MAPPING. Multiple transformations can be combined to yield a single transformation which should have the same effect as the sequential application of original ones. CONCATENATION /kənˌkatnˈāSH(ə)n/ Equation of P* for graphics hardware should be in matrix notation: P* = [T]P, where [T] is the transformation matrix.

Содержание слайда: Translation Translation is a rigid-body transformation (Euclidean) when each entity of the model remains parallel, or each point moves an equal distance in a given direction: P* = P + d (for both 2D and 3D). In a scalar form (for 3D): x* = x + xd y* = y + yd z* = z + zd

Содержание слайда: Scaling Scaling is used to change the size of an entity or a model. P* = [S]P sx 0 0 For general case [S] = 0 sy 0 , 0 0 sz If 0 < s < 1 - compression If s > 1 - stretching sx = sy = sz - uniform scaling, otherwise - non-uniform

Содержание слайда: Mirror Plane* => Negate the corresponding coordinate Mirror through Line* => Reflect through 2 planes intersecting at the axis Point* => Reflect through 3 planes intersecting at the point * plane - principal plane, line - X, Y, or Z axes, point - CS origin P* = [M]P, where [M] = = Question: Define the signs (in the matrix) for the reflections (mirroring) through: a) x = 0, y = 0, z = 0 planes b) X, Y, and Z axes c) the CS origin

Содержание слайда: Rotation Rotation is a non-commutative transformation (depends on sequence).

Содержание слайда: Homogeneous Transformation - 1 When we scale then rotate, the transformed image is given by: P* = ([R][S])P where [S], [R], [R] [S] are 3x3 transformation matrices. This is not the case for a translation (P* = P + d). The goal is to find a [D] such that P + d = [D]P in order to perform valid matrix multiplication. This is found by using a homogeneous coordinates. Homogeneous Transformation maps n-dimensional space into (n+1)- dim. 3D representation of the point vector - P = [x, y, z]T Homogeneous rep. of the same vector - P = [xw, yw, zw, w]T where w = 1

Содержание слайда: Homogeneous Transformation - 2 The transformation matrices in new (homogeneous) representation:

Содержание слайда: Composition of Transformations Now we are able to include all the transformations in a single matrix. In case of composition of transformations: P* = [Tn][Tn-1]...[T2][T1]P, where [Ti] are different transformation matrices. Sequence is important! Practice: Mirror point A through the given line and find x and y.

Содержание слайда: Another example Scale line AB about point M by factor of 2 and then mirror new line A’B’ about the origin.