IITD CSE Summer Workshop Computer Graphics

IITD CSE Summer WorkshopComputer Graphics Subodh Kumar

1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 x y z 1 Perspective Transformation x,y,z,w x,y,z,z 1

Linear and Affine transform • Linear: x’ = Ax • Scale, Rotation • T(ax + by) = aT(x) + bT(y) • Affine: Translate • x’ = Ax + b • Transforms a line to a line • Respects parallelism • Does not respect angles and lengths

Projective Transformation • n-dimensional projective space is an n+1-dimensional vector space • 0,0,0,… is not a part of this space • x = kx • xn+1 == 0 => Point at infinity {= Direction} • Actually any coordinate may be chosen • Affine space is a projection of Projective space • Typically on the plane xn+1 = 1 • Projective transformation • xj = ∑aixi • Ax, A is a 4x4 matrix (15 DOF!)

Normal Transformation n.p = 0 nTp = 0 => n’T Mp must be 0 => (M’n)T Mp = 0 => nTM’T Mp = 0 => M’T M = kI => M’T = M-1 (k = 1) => M’ = k(M-1)T

zp-zl xp-xl = zr-zl xr-xl Attribute Interpolation x0,y0,z0 y = mx + c x = 1/m(y+c) =ay+b x2,y2,z2 x+1 = ay+a + b = x + a x1, y1,z1 zp+1 = zp + k

Perspective Distortion z2 z1

Perspective Distortion

Perspective Correction • Linear interpolation: • p(t) = p1 + (p2 – p1) u, 0 <= u<= 1 • Screen space interpolation: • ys = y1s + t (y2s – y1s) • same as: y/z = y1/z1 + t (y2/z2 – y1/z1) • But really: • y = y1 + u (y2 – y1) • z = z1 + u (z2 – z1)

Perspective Correction

Perspective Corrected

Xlib GLX OpenGL GLU Application GDI WGL OGL/DX GLU Application Graphics SW ArchitectureTop View Windows Unix

OpenGL Architecture Client State Update / Action Pushbuffer Machine state Frame Buffer Must Respect PB order

IITD CSE Summer Workshop Computer Graphics