Viewing in 3D Viewing in 3D Foley & Van Dam, Chapter 6 • Transformation Pipeline • Viewing Plane • Viewing Coordinate System • Projections • Orthographic • Perspective
OpenGL Transformation Pipeline Viewing Coordinate System Homogeneous coordinates in World System zw world
yw ModelViewModelView Matrix Matrix xw Tractor Viewing System Viewer Coordinates System ProjectionProjection Matrix Matrix Clip y Coordinates v Front- xv ClippingClipping Wheel
System P0 zv ViewportViewport Transformation Transformation ne pla ing Window Coordinates View
Specifying the Viewing Coordinates Specifying the Viewing Coordinates
• Viewing Coordinates system, [xv, yv, zv], describes 3D objects with respect to a viewer zw y v P v xv •A viewing plane (projection plane) is set up N P0 zv perpendicular to zv and aligned with (xv,yv) yw xw ne pla ing • In order to specify a viewing plane we have View to specify: •P0=(x0,y0,z0) is the point where a camera is located •a vector N normal to the plane • P is a point to look-at •N=(P-P)/|P -P| is the view-plane normal vector •a viewing-up vector V 0 0 •V=zw is the view up vector, whose projection onto • a point on the viewing plane the view-plane is directed up Viewing Coordinate System Projections V u N z N ; x ; y z u x • Viewing 3D objects on a 2D display requires a v v V u N v v v mapping from 3D to 2D • The transformation M, from world-coordinate into viewing-coordinates is: • A projection is formed by the intersection of certain lines (projectors) with the view plane 1 2 3 ª x v x v x v 0 º ª 1 0 0 x 0 º « » « » y 1 y 2 y 3 0 0 1 0 y • Projectors are lines from the center of projection M « v v v » « 0 » R T 1 2 3 « z z z 0 » « 0 0 1 z 0 » « v v v » « » through each point in the object ¬« 0 0 0 1 ¼» ¬ 0 0 0 1 ¼ Center of • Defining the camera in OpenGL: Projection glMatrixMode(GL_MODELVIEW); glLoadIdentity();
gluLookAt(P0x, P0y, P0z, Px, Py, Pz, Vx, Vy, Vz);
Projections Projections • Center of projection at infinity results with a parallel • Parallel projections preserve relative proportions of projection objects, but do not give realistic appearance (commonly used in engineering drawing) • A finite center of projection results with a perspective projection • Perspective projections produce realistic appearance, but do not preserve relative proportions
Perspective Projection
Parallel Projection Orthographic Projection
• Projectors are all parallel Since the viewing plane is aligned with (xv,yv), • Orthographic: Projectors are perpendicular orthographic projection is performed by: to the projection plane ª x p º ª x v º ª 1 0 0 0 º ª x v º « y » « y » « 0 1 0 0 » « y » • Oblique: Projectors are not necessarily « p » « v » « » « v » « 0 » « 0 » « 0 0 0 0 » « z v » perpendicular to the projection plane « » « » « » « » ¬ 1 ¼ ¬ 1 ¼ ¬ 0 0 0 1 ¼ ¬ 1 ¼
Orthographic Oblique
(x,y,z) (x,y)
yv xv
P0 zv Orthographic Projection Oblique Projection • Lengths and angles of faces parallel to the • Projectors are not perpendicular to the viewing planes are preserved viewing plane • Problem: 3D nature of projected objects is • Angles and lengths are preserved for faces parallel to the plane of projection difficult to deduce Top View • Preserves 3D nature of an object
w Fr ie on e V t v id yv (x ,y ) iew S p p xv (x,y,z) (x,y)
Oblique Projection Oblique Projection
•Two types of oblique projections are ª x p º ª 1 0 a cos I 0 º ª x v º ª x v z v a cos I º « » « » « » « » y p 0 1 a sin I 0 y v y v z v a sin I commonly used: « » « » « » « » « 0 » « 0 0 0 0 » « z v » « 0 » – Cavalier: R tan « » « » « » « » D ¬ 1 ¼ ¬ 0 0 0 1 ¼ ¬ 1 ¼ ¬ 1 ¼ – Cabinet: D tan-1 |R
y yv v 1/a=tan(D) (x ,y ) (xp,yp) z/b= 1/a p p b=za D D b b x =z a cos( ) p I x a xv a v (x,y,z) I yp=zasin(I) (x,y,z) I (x,y,1) (x,y,1) (x,y) (x,y)
Oblique Projection Oblique Projection • Cavalier projection : 1 1 1 1 – Preserves lengths of lines perpendicular 1 1 to the viewing plane I=45o I=30o – 3D nature can be captured but shape seems distorted Cavalier Projections of a cube – Can display a combination of front, side, for two values of angle I and top views • Cabinet projection: 0.5 1 1 0.5 – Lines perpendicular to the viewing plane project 1 1 at 1/2 of their length I=45o I=30o – A more realistic view than the Cavalier projection Cabinet Projections of a cube for two values of angle I – Can display a combination of front, side, and top views Perspective Projection Perspective Projection • In a perspective projection, the center of projection is at a finite distance from the viewing plane • The size of a projected object is inversely proportional to it distance from the viewing plane • Parallel lines that are not parallel to the viewing plane, converge to a vanishing point • A vanishing point is the projection of a point at infinite distance Z-axis y vanishing point
x
z
Vanishing Points Vanishing Points • There are infinitely many general vanishing y points • There can be up to three principal vanishing x points (axis vanishing points) • Perspective projections are categorized by z the number of principal vanishing points, equal One point (z axis) perspective projection to the number of principal axes intersected by x axis z axis the viewing plane vanishing point vanishing point • Most commonly used: one-point and two- points perspective Two points perspective projection
Perspective Projection Perspective Projection Thus, a perspective projection matrix is defined as: (x,y,z) (x ,y ,0) p p ª 1 0 0 0 º « 0 1 0 0 » y « » d M center of x per « 0 0 0 0 » « 1 » projection x (x,y,z) « 0 0 1 » ¬ d ¼ z xp d z ª 1 0 0 0 º ª x º ª x º « 0 1 0 0 » « » « y » « » y « » • Using similar triangles it follows: M P « » per « 0 0 0 0 » « z » « 0 » « 1 » « » « z d » x p x y p y « 0 0 1 » 1 « » ; ¬ d ¼ ¬ ¼ ¬ d ¼ d z d d z d d x d y d x d y x ; y ; z 0 x p ; y p ; z p 0 p p p z d z d z d z d Perspective Projection Projections What is the difference between moving the center of •M is singular (|M |=0), thus M is a many to per per per projection and moving the projection plane? one mapping (for example: M P=M 2P) per per Original • Points on the viewing plane (z=0) do not change
Center of Projection z • The homogeneous coordinates of a point at infinity Projection plane Moving the Center of Projection directed to (Ux,Uy,Uz) are (Ux,Uy,Uz,0). Thus, The vanishing point of parallel lines directed to (Ux,Uy,Uz) is at [dUx/Uz, dUy/Uz] Center of Projection z Projection plane Moving the Projection Plane • When dof, Mper oMort
Center of Projection z Projection plane
Projections
Planar geometric projections
Parallel Perspective
Oblique Orthographic One point Front Top Two point Cavalier Side Other Cabinet Other Three point