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Lecture 2 - Transformations for animation (slides)

# a point and a direction graphics lecture 2 slide

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Unformatted text preview: ight in with deﬁnition of straight line in ininwith deﬁnition which uses line Cartesian space of straight Ties Cartesian space which a lineusesCartesian space which uses point and a direction in a point and a direction ! ! a point and a direction Graphics Lecture 2: Slide 26 ! 25 / 45 Adding two position vectors Adding two position vectors ! • If add two position vectors we obtain their their mid-point ! If  wewe add two position vectors we obtainmid-point 01010 10 1 Xa Xb Xa + Xb (Xa + Xb ) / 2 B Ya C B Yb C B Ya + Yb C B (Ya + Yb ) / 2 C B C+B C=B C=B C @ Za A @ Yb A @ Za + Zb A @ (Za + Zb ) / 2 A 1 1 2 1 •  This is reasonable since adding two position vectors has This is reasonable since adding two position vectors has no real no real meaning in vector geometry ! meaning in vector geometry Graphics Lecture 2: Slide 27 ! 26 / 45 The structure of a transformation matrix The structure of a transformation matrix ! The bottom row is always 0 0 0 1 •  The bottom row is always 0 0 0 1 ! • The columns of a transformation matrix three direction The  columns of a transformation matrix comprisecomprise three direction vectors and one vectors and one position vector position vector ! Matrix Matrix ! 0 1 qx rx sx Cx Bqy ry sy Cy C B C @qz rz sz Cz A 0001 Direction Direction vectors ! vectors 01 qx B qy C BC @ qz A 0 01 rx B ry C BC @ rz A 0 01 sx Bsy C BC @sz A 0 Position Position vectors vector! 01 Cx BCy C BC @Cz A 1 Graphics Lecture 2: Slide 28 ! 27 / 45 Characteristics of Transformation matrices Direction vector: CharacteristicsTransformation matrices a↵ected ! of transformation not Characteristics ofZero, in the last ordinate ) matrices by the translation. Characteristics of Transformation matrices 0 10 1 0 1 Direction vector: Zero,rx the last ordinate ) not a↵ected by the qx in sx Cx ⇤ ⇤ Direction vector: Zero, in the last ordinate ⇒ ) not a↵ected by the Zero, in the last ordinate not affected by the •translation.   Direction vector:0q B y ry sy Cy C B⇤C B⇤C 10 1 0 1 B CB C B C translation. ! 0 qx rx sx Cx1 0 ⇤1 = 0 ⇤1 translation. @q z r z s...
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