Matrices of Projection Methods
Centre of projection S = (a, b, c, 1),
for c 0, ( or b 0)
is a real point in the space ,
not located in the image plane - p = xy ( or n = xz ) .
View of an arbitrary point A in the space , A S is the point AS in the image plane,
which is the intersection point of the projecting line sA = AS passing through the point A
and the centre of projection S with the image plane
AS = sA Ç p ( or AS = sA Ç n )Distance of the centre of projection to the image plane is d = | Sp | = c ( or d = | Sn | = b ) .
Matrix of the central projection to the ground image plane p = xy
AS = A . PSp =
Matrix of the central projection to the frontal image plane n = xz
AS = A . PSn =
Centre of projection S has no view.
Centre of projection S = (0, 0, -d, 1), distance d = 30cm is the distance of the eye to the image plane.
Matrix of the linear perspective to the ground image plane p = xy
Stereoscopic projection with two centres
Centres of projection 1S = (a, b, -d, 1), 2S = (-a, b, -d, 1),
distance of the eye to the image plane d = 30cm,
distance of centres |1S 2S | = 2b = 6,5cm is the average distance of eyes
Direction of projection s - a pencil of lines in the space
in the same direction, sharing a commmon ideal point with the line s,
determined by the direction vector s = (a, b, c, 0), c 0, ( or b 0 ),
intersects image plane p ( or n ) in the real point.
View of an arbitrary point A in the space is the point AP Î p ( or AP Î n ),
which is the intersection point of the line sA, A Î sA in the direction s passing through the point A with the image plane
AP = sA Ç p ( or AP = sA Ç n ) .
Matrix of the parallel projection to the ground image plane p = xy
AR = A . PRp =
Matrix of the parallel projection to the frontal image plane n = xz
AR = A . PRn =
Orthographic projection to plane p = xy is represented by the matrix
with two elective angles the azimuth a and the elevation e from the interval <-360°, 360°>.
Monge Method (Multiview Orthographic Drawing)
|GROUND VIEW||FRONT VIEW||SIDE VIEW|
Orthogonal Axonometry ( a, e 0°, ±90°, ±180°, ±270°, ±360° )
|AXONOMETRIC VIEW||AXON. GROUND VIEW|
|AXON. FRONT VIEW||AXON. SIDE VIEW|