**Matrices of Projection Methods**

**Central Projection**

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 *A*_{S} in the image plane,

which is the intersection point of the projecting line *s ^{A}* =

and the centre of projection

*AS* = *s ^{A}* Ç p ( or

Matrix of the central projection to the ground image plane p = *xy*

** A_{S} = A . P_{Sp} = **

Matrix of the central projection to the frontal image plane n = *xz*

** A_{S} = A . P_{Sn} = **

Centre of projection *S* has no view.

**Linear perspective**

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 ^{1}*S* = (*a*, *b*, -*d*, 1), ^{2}*S* = (-*a*, *b*, -*d*, 1),

distance of the eye to the image plane *d* = 30cm,

distance of centres |^{1}*S* ^{2}*S* | = 2*b* = 6,5cm is the average distance of eyes

**Parallel Projection**

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
*A _{P}* Î p ( or

which is the intersection point of the line

*A _{P}* =

Matrix of the parallel projection to the ground image plane p = *xy*

*A _{R}* =

Matrix of the parallel projection to the frontal image plane n = *xz*

** A_{R} = A . P_{Rn} = **

Orthographic projection to plane p = *xy* is represented by the matrix

**P**

with two elective angles the **azimuth** a and the **elevation** e from the interval <-360°, 360°>.

1. a = e = 0° front view (from the front)

2. a = 0°, e = 90° ground view (from the top)

3. a = 90°, e = 0° side view (from the left side)

4. a = -90°, e = 0° side view (from the right side)

5. a = 0°, e e = -90° bottom view (from the bottom)

6. a = 180°, e = 90° back view( from the backward)

7. a = ±45°, e = ±45° isometry (obr. 1.47a)

8. a = 45°, e 0°, ±45°, ±90°, ±135°, ±180° , ±225° , ±270° , ±315°, ±360° dimetry (obr. 1.47b)

9. a , e 0°, ±45°, ±90°, ±135°, ±180° , ±225° , ±270° , ±315°, ±360° trimetry (obr. 1.47c)

**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 |
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AXON. FRONT VIEW | AXON. SIDE VIEW |
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