Basic Projections 2D to 3D

3D viewing process Specify a 3D view volume Clip against view volume Project onto a 2D viewing plane Define a window on the viewing plane Apply 2D viewing transformations to map window contents into 2D-image viewport

Planar Projections •Perspective: Distance to CoP is finite •Parallel: Distance to CoP is infinite

Parallel Projections •Orthographic: Direction of projection is orthogonal to the projection plane •Elevations: Projection plane is perpendicular to a principle axis •Front •Top (Plan) •Side •Axonometric: Projection plane is not orthogonal to a principle axis •Isometric: Direction of projection makes equal angles with each principle axis. •Oblique: Direction of projection is not orthogonal to the projection plane; projection plane is normal to a principle axis •Cavalier: Direction of projection makes a 45° angle with the projection plane •Cabinet: Direction of projection makes a 63.4° angle with the projection plane

Perspective Projections •One-point: One principle axis cut by projection plane One axis vanishing point •Two-point: Two principle axes cut by projection plane Two axis vanishing points •Three-point: Three principle axes cut by projection plane Three axis vanishing points

Perspective Projections

One- Point Projections Center of Projection on the negative z-axis View-plane parallel to the x-y plane and through the origin. -Z +Z (x, y, z) (xproj, yproj, 0) x y xprojected = xd/(d+z) = x/(1+(z/d)) yprojected = yd/(d+z) = y/(1+(z/d) (1 0 0 0) (x) (x) (0 1 0 0) (y) = (y) (0 0 0 0) (z) (0) (0 0 1/d 1) (1) (z/d + 1) (0, 0, -d)

One- Point Projections (x, y, z) Center of Projection at the origin viewplane parallel to the x-y plane a distance d from the origin. xprojected = dx/z = x/(z/d) yprojected = dy/z = y/(z/d) (xproj, yproj, d) y x (1 0 0 0) (x) (x) (0 1 0 0) (y) = (y) (0 0 1 0) (z) (z) (0 0 1/d 0) (1) (z/d) Mper Points plotted are x/w, y/w where w = z/d