“ Projection can be defined as a mapping of point P(x,y,z) onto its image P’(x’,y’,z’) in the projection plane, which constitute the display surface”. The mapping is determind by a projection line called the “Projector” that passes through P and intersect the view plane. The intersection point is P’.
“ The techniques of projection are generalizations of the principles used by artists in preparing perspective drawing of 3D objects and scenes. The eye of the artist is placed at the center of projection, and the canvas (view plane).”
A Perspective projection is determined by prescribing a center of projection and a view plane. The view plane is determined by its view reference point R o and view plane normal N. the object point P is located in world co-ordinates at (x,y,z). The problem is to determine the image point co-ordinates P’(x’,y’,z’).
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View Plane Center of projection View Plane P 1 (x,y,z) P 2 P’ 1 (x’,y’,z’) Z Y X C N View Plane Normal
These points are formed by the intersection of lines parallel to one of the three principal axis. The number of principal vanishing points is determined by the number of principal axis intersected by the view plane.
One Principal Vanishing Point Perspective Projection
Two Principal Vanishing Point Perspective Projection
Three Principal Vanishing Point Perspective Projection
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One Principal Vanishing Point Perspective Projection
This Perspective projection occurs when the projection plane is perpendicular to one of the Principal axis (x or y or z).
Assume that it is z-axis. The view plane normal vector N is the vector N 1 an Principal Vanishing Point is
P 3 X 3 =a 1 Y 3 =b 1 Z 3 =c 1 + d 1 n 3 : Therefore, V=V 1 I+V 2 J+V 3 K X=V 1 t+l Y=V 2 t+m Z=V 3 t +n P(l,m,n) K=N.V=n 1 v 1 +n 2 v 2 +n 3 v 3 Where a 1 ,b 1 ,c 1 are coordinates of the centre of projection n 1 ,n 2 ,n 3 are components of view plane normal vector d 1 is proportional to distance D from the view plane to the Centre of Projection
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Two Principal Vanishing Point Perspective Projection
“ This Perspective projection occurs when the projection plane intersects exactly two of the principal axes.”
“ The parallel projection used by drafters and engineers to create working drawings of an object which preserves its scale and shape. The complete representation of these details often requires two or more views (projections) of the object onto different view planes.
In parallel projection, image points are found as the intersection of the view plane with a projector drawn from the object point and having a fixed direction.
P(x,y,z) P 2 P’(x’,y’,z’) P 2 ’ X Z Y Direction of V Projection View Plane N
A Parallel projection is determined by prescribing a direction of projection vector V and a view plane. The view plane is determined by its view reference point R o and view plane normal N. the object point P is located in world co-ordinates at (x,y,z). The problem is to determine the image point co-ordinates P’(x’,y’,z’).
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