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Three-Dimensional Graphics

Three-Dimensional Graphics. A 3D point (x,y,z) – x,y, and Z coordinates We will still use column vectors to represent points Homogeneous coordinates of a 3D point (x,y,z,1) Transformation will be performed using 4x4 matrix. T. y. x. z. Right hand coordinate system.

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Three-Dimensional Graphics

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  1. Three-Dimensional Graphics • A 3D point (x,y,z) – x,y, and Z coordinates • We will still use column vectors to represent points • Homogeneous coordinates of a 3D point (x,y,z,1) • Transformation will be performed using 4x4 matrix T y x z

  2. Right hand coordinate system • Xx Y = Z ; Y x Z = X; Z x X = Y; Y y z x +z x Left hand coordinate system Right hand coordinate system

  3. 3D transformation • Very similar to 2D transformation • Translation x’ = x + tx; y’ = y + ty; z’ = z + tz X’ 1 0 0 tx X Y’ 0 1 0 ty Y Z’ 0 0 1 tz Z 1 0 0 0 1 1 • OpenGL - glTranslated(tx, ty, tz); = homogeneous coordinates

  4. 3D transformation • Scaling X’ = X * Sx; Y’ = Y * Sy; Z’ = Z * Sz X’ Sx 0 0 0 X Y’ 0 Sy 0 0 Y Z’ 0 0 Sz 0 Z 1 0 0 0 1 1 • OpenGL - glScaled(Sx, Sy, Sz); =

  5. y + x z 3D transformation • 3D rotation is done around a rotation axis • Fundamental rotations – rotate about x, y, or z axes • Counter-clockwise rotation is referred to as a positive rotation (when you look down negative axis)

  6. 3D transformation • Rotation about Z – similar to 2D rotation x’ = x cos(q) – y sin(q) y’ = x sin(q) + y cos(q) z’ = z y x + cos(q) -sin(q) 0 0 sin(q) cos(q) 0 0 0 0 1 0 0 0 0 1 z • OpenGL - glRotatef(q, 0,0,1)

  7. x y x z + + z y 3D transformation • Rotation about y z’ = z cos(q) – x sin(q) x’ = z sin(q) + x cos(q) y’ = y cos(q) 0 sin(q) 0 0 1 0 0 -sin(q) 0 cos(q) 0 0 0 0 1 • OpenGL - glRotatef(q, 0,1,0)

  8. z y x y + + z x 3D transformation • Rotation about x y’ = y cos(q) – z sin(q) z’ = y sin(q) + z cos(q) x’ = x 1 0 0 0 0 cos(q) -sin(q) 0 0 sin(q) cos(q) 0 0 0 0 1 • OpenGL - glRotatef(q, 1,0,0)

  9. (rx, ry, rz) y x z 3D transformation • Arbitrary rotation axis (rx,ry,rz) • Text pp. 193 explains how to do it • We omit the detail here • Use OpenGL: glRotatef(angle, rx, ry, rz) • Can fill in all 9 entriesof the rotation matrix.

  10. OpenGL Transformation Composition • A global modeling transformation matrix (GL_MODELVIEW, called it M here) glMatrixMode(GL_MODELVIEW) • The useris responsible to reset it if necessary glLoadIdentity() -> M = 1 0 0 0 1 0 0 0 1

  11. OpenGL Transformation Composition • Matrices for performing user-specified transformations are multiplied to the current matrix • For example, 1 0 1 glTranslated(1,1 0); M = M x 0 1 1 0 0 1 • All the vertices defined within glBegin() / glEnd() will first go through the transformation (modeling transformation) P’ = M x P

  12. Object Local Coordinates Object World Coordinates Transformation Pipeline Modeling transformation …

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