3D Viewing. Chapter 7. Projections. 3D clipping. OpenGL viewing functions and clipping planes

Transcription

1 3D Viewing Chapter 7 Projections 3D clipping OpenGL viewing functions and clipping planes 1

2 Projections Parallel Perspective Coordinates are transformed along parallel lines Relative sizes are preserved Parallel lines remain parallel Projection lines converge at a point More distant objects are relatively smaller Closer to the way we actually see things 2

3 Orthogonal Projections An orthogonal projection is a parallel projection where the transformation lines are perpendicular to the view plane. Orthogonal projections are used for engineering drawings front and side elevations and plan view. Axonometric projection lets you see multiple faces of an object at once. An isometric projection has the view direction along a (1,1,1) direction of the world coordinate system. 3

4 Orthogonal Projections If the view plane is perpendicular to the z-axis, x p = y p = x y The z coordinate is used only for visibility calculations. 4

5 Clipping The view volume for an orthogonal projection is a rectangular parallelepiped with a cross-section the same size as the clipping window. This will be normalized to a cube. 5

6 Normalization The normalization transformation involves translation and scaling in all 3 directions. 2 xwmax xw min 0 0 xw max+xw min xw max xw min 2 0 xwmax xw M ortho, norm min 0 xw max+xw min xw max xw min 2 z 0 0 near +z far znear z far z near z far

7 Oblique Parallel Projections For this kind of projection, projection paths are not perpendicular to the view plane. Use two angles to specify the projection. Point (x, y, z) projects to (xp, y p, z vp ). xp = yp = x + L cos φ y + L sin φ 7

8 Oblique Parallel Projections the projection line makes an angle α with the line between the projected point and the orthogonal projection point. tan α = zvp z L An orthogonal projection has α = 90. Oblique projections correspond to a shearing transformation. Can use a projection vector to specify the projection. x 8

9 Cabinet and Cavalier Projections Typical choices for φ are 30 and 45. A cavalier projection has α = 45 A cabinet projection has tan α = 2 so α

10 Oblique Transformations The view volume is a parallelepiped. Use a shear transformation to convert this volume into a rectangular parallelepiped. 1 0 V px V V pz z px vp V pz 0 1 V py V V Moblique = pz z py vp V pz Then transform as for an orthogonal projection. This is applied after MW C, V C. Moblique, norm = M ortho,norm M oblique 10

11 Perspective Projections Parallel projections are easy but not very realistic looking. Perspective transformations are more realistic. Points are projected along lines that meet at a reference point. P 1 Projection Reference Point y view (x prp, y prp, z prp ) P 2 z view More distant objects appear relatively smaller in a perspective projection. 11

12 Transformations for Perspective Projections Use parametric equations for positions along a projection line. x = x (x x prp )u y = y (y y prp )u z = z (z z prp )u At (x, y, z), u = 0 and at (x prp, y prp, z prp ), u = 1. If the view plane is at z = zvp, then u = z vp z z prp z Then, x and y can be calculated from u. Graphics packages often restrict the projection point to either be at the origin or along the zview axis which makes the expressions for x and y simpler. 12

13 Vanishing Points In a perspective projection, parallel lines that aren t parallel to the view point converge at a vanishing point. Each direction has its own vanishing point; principal vanishing points are for lines parallel to one of the coordinate axes. Depending on the orientation of the view plane, there may be 1, 2 or 3 principal vanishing points. 13

14 View Volumes The view volume for perspective projections is an infinite pyramid formed from lines starting at the perspective reference point and going through the corners of the view plane. Adding near and far clipping planes reduces this to a frustum. 14

15 Perspective Transformations The values of xp and y p calculated earlier have coefficients that depend on z. Need to use a transformation to homogeneous coordinates of the form x p = y p = x(z prp z vp )+x prp (z vp z) z prp z y(z prp z vp )+y prp (z vp z) z prp z This converts the description to homogeneous parallel-projection coordinates. 15

16 Viewport Transformation A perspective transformation can be either symmetric or oblique. Use a shear transformation to convert an oblique perspective to a symmetric one. 16

17 OpenGL Viewing Functions Specify viewing parameters with glulookat( x0, y0, z0, xref, yref, zref, vx, vy, vx) where P 0 = (x 0, y 0, z 0 ) is the viewing origin P ref = (x ref, y ref, z ref ) is the look-at point V = (V x, v y, V z ) is the view-up vector The positive zview-axis is in the direction N = P 0 P ref Default parameters are P 0 = (x 0, y 0, z 0 ) = (0, 0, 0) P ref = (0, 0, 1) V = (0, 1, 0) 17

18 Orthogonal Projections An orthogonal projection is specified with glmatrixmode( GLPROJECTION); glortho( xwmin, xwmax, ywmin, ywmax, dnear, dfar); Default parameters are xwmin = ywmin = dnear = -1 xwmax = ywmax = dfar = 1 Note that znear is behind the viewing position. gluortho2d is equivalent to gluortho with dnear = -1.0 and dfar = 1.0. To do oblique parallel projections you need to either rotate the scene to get the effect you want or set up the matrices manually. 18

19 Field of View 19

20 Perspective Transformations For a symmetric perspective transformation, use gluperspective( theta, aspect, dnear, dfar) For a general perspective transformation, use glfrustum( xwmin, xwmax, ywmin, ywmax, dnear, dfar); 20

21 3D Clipping Algorithms Often do the clipping after normalization to make it more efficient. This also allows clipping to be done in hardware. Several algorithms Region Clipping Polygon clipping Arbitrary Clip planes 21

22 3D Region Codes Now have clipping planes instead of lines. Extend the region codes to have two extra bits. bit 6 bit 5 bit 4 bit 3 bit 2 bit 1 Far Near Top Bottom Right Left 22

23 3D Region Codes Then use the same basic rules as in the 2D algorithm for elimination and calculating intersections

24 Polygon Clipping Since most graphics packages deal with objects constructed from polygons, clipping is basically a problem of clipping polygons in 3D. Coordinate extents (bounding boxes) can be used for trivial rejection or acceptance. Then clip edges to modify and create new vertex lists. 24

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26 OpenGL Clipping Planes Sometimes need to specify additional clipping planes. Use glclipplane( id, planeparameters); glenable( id); The values of id are GLCLIP_PLANE0, GL_CLIP_PLANE1 The plane parameters are given as a vector with the 4 coefficients from the equation of the plane. 26