Lecture 11: 2D Transformations I - Vectors and Vector Operations, Slides of Computer Graphics

Download Lecture 11: 2D Transformations I - Vectors and Vector Operations and more Slides Computer Graphics in PDF only on Docsity! Computer Graphics Lecture 11 2D Transformations I Docsity.com Vectors Another important mathematical concept used in graphics is the Vector  If P1 = (x1, y1, z1) is the starting point and P2 = (x2, y2, z2) is the ending point, then the vector V = (x2 – x1, y2 – y1, z2 – z1) Docsity.com Vector Projections y eee Eee » Projection of v onto the re D CES BYere-) Wee) At] Projection of v onto the xz plane ® BYere-) Wee) At] 2D Magnitude and Direction  The magnitude (length) of a vector: |V| = sqrt( Vx 2 + Vy 2 ) • Derived from the Pythagorean theorem  The direction of a vector: a = tan-1 (Vy / Vx) a is angular displacement from the x-axis Docsity.com Vector Normalization  “Normalizing” a vector means shrinking or stretching it so its magnitude is 1 –creating unit vectors –Note that this does not change the direction Normalize by dividing by its magnitude; V = (1, 2, 3) |V| = sqrt( 1 2 + 2 2 + 3 2 ) = sqrt(14) = 3.74 Docsity.com Vector Normalization Vnorm = V / |V| = (1, 2, 3) / 3.74 = (1 / 3.74, 2 / 3.74, 3 / 3.74) = (.27, .53, .80) |Vnorm| = sqrt( .27 2 + .53 2 + .80 2 ) = sqrt( .9 ) = .95  Note that the last calculation doesn’t come out to exactly 1. This is because of the error introduced by using only 2 decimal places in the calculations above. Docsity.com Vector Addition Equation: V3 = V1 + V2 = (V1x+V2x , V1y+V2y , V1z+V2z) Visually: Docsity.com Why is dot product important for graphics? – It is zero iff the 2 vectors are perpendicular • cos(90) = 0 Docsity.com  The Dot Product computation can be simplified when it is known that the vectors are unit vectors V1 . V2 = cos(q) because |V1| and |V2| are both 1 – Saves 6 squares, 4 additions, and 2 square roots Docsity.com Cross Product  The cross product of 2 vectors is a vector V1 x V2 = ( V1y V2z - V1z V2y , V1z V2x - V1x V2z , V1x V2y - V1y V2x ) – Note that if you are into linear algebra there is also a way to do the cross product calculation using matrices and determinants Docsity.com Forming Coordinate Systems  Cross products are great for forming coordinate system frames (3 vectors that are perpendicular to each other) from 2 random vectors – Cross V1 and V2 to form V3 – V3 is now perpendicular to both V1 and V2 – Cross V2 and V3 to form V4 – V4 is now perpendicular to both V2 and V3 – Then V2, V4, and V3 form your new frame Docsity.com V1 and V2 are in the new xy plane Docsity.com Basic Transformations  Translation  Rotation  Scaling Docsity.com Translation Distances tx, ty For point P(x,y) after translation we have P′(x′,y′) x′ = x + tx , y′ = y + ty  (tx, ty) is Translation vector Docsity.com Now x′ = x + tx , y′ = y + ty can be expressed as a single matrix equation: P′ = P + T Where P = P′ = T =       y x       ' ' y x       y x t t Docsity.com Rigid Body Transformation Objects are moved without deformation Every point on the object is translated by the same amount Typically all the endpoints are translated and object is redrawn using new endpoint positions Docsity.com For simplicity, consider the pivot at origin and rotate point P (x,y) where x = r cosФ and y = r sinФ  If rotated by θ then:  x′ = r cos(Ф + θ) = r cosФ cosθ – r sinФ sinθ and  y′ = r sin(Ф + θ) = r cosФ sinθ + r sinФ cosθ Docsity.com Replacing r cosФ with x and r sinФ with y, we have: x′ = x cosθ – y sinθ and y′ = x sinθ + y cosθ Docsity.com Column vector representation: P′ = R . P Where         qq qq cossin sincos R        y x P        ' ' ' y x P Docsity.com Rigid Body Transformation Objects are rotated without deformation Every point on the object is rotated by the same angle Typically all the endpoints are rotated and object is redrawn using new endpoint positions Docsity.com Scaling Y X 0 1 1 2 2 3 4 5 6 7 8 9 10 3 4 5 6       1 2       1 3       2 6       2 9 2 3   y x s s Docsity.com Scaling Scaling alters the size of an object Scaling factors Sx and Sy are used For polygons, coordinates of each vertex may be multiplied by Sx & Sy to produce the transformed coordinates: x′ = x.Sx y′ = y.Sy Docsity.com When scaling w.r.t. origin, both the size and distance form origin get scaled. Size and distance fro origin: for Scaling factor 2 (double the size and double the distance from origin.) for Scaling factor 0.5 (reduce the size to 50% and also the distance from origin to 50%. Docsity.com This can be controlled by scaling w.r.t. a fixed point (xf,yf) • x’ = xf + (x - xf)Sx • y’ = yf + (y - yf)Sy  These can be rewritten as: • x’ = x. Sx + xf (1 – Sx) • y’ = y. Sy + yf (1 – Sy) • Where the terms xf (1 – Sx) and yf (1 – Sy) are constant for all points in the object Docsity.com