How to construct an orthogonal projection

How to construct an orthogonal projection

Orthogonal, or rectangular, a projection (Latin proectio − "throwing forward") can be presented physically as the shadow cast by a figure. When designing buildings and other objects the projective image is also used.


1. To receive a point projection to an axis, construct a perpendicular to an axis of this point. The perpendicular basis (point in which the perpendicular crosses a projection axis) and will be, by definition, required size. If the point on the plane has coordinates (x, y), its projection to an axis of Ox will have coordinates (x, 0), on Oy axis − (0, y).

2. Let now on the plane the piece be set. To find its projection to a coordinate axis, it is necessary to restore perpendiculars to an axis from its extreme points. The turned-out piece on an axis will also be an orthogonal projection of this piece. If trailer points of a piece had the coordinate (A1,B1) and (A2,B2), then its projection to an axis of Ox will be located between points (A1.0) and (A2.0). Extreme points of a projection to an axis of Oy will become (0, B1), (0, B2).

3. For creation of a rectangular projection of a figure to an axis carry out perpendiculars from extreme points of a figure. For example, the piece equal to diameter will be a circle projection to any axis.

4. To receive an orthogonal projection of a vector to an axis, construct projections of the beginning and the end of a vector. If the vector is already perpendicular to a coordinate axis, its projection degenerates in a point. Like a point the zero vector which does not have length is projected. If free vectors are equal, then also their projections are equal.

5. Let the vector of b form with axis x a corner ψ. Then vector projection to an axis of Pr of xb = |b| · cosψ. For the proof of this situation consider two cases: when corner ψ sharp and stupid. Use definition of a cosine, finding it the relation of an adjacent leg to a hypotenuse.

6. Considering algebraic properties of a vector and its projections, it is possible to notice that: 1) The projection of the sum of vectors of a+b is equal to the sum of projections of Pr of xa+ Pr of xb; 2) The projection of a vector of b increased by Q scalar is equal to a projection of a vector of b increased by the same number Q: Ave. (x) Qb=Q · Ave. (x) b.

7. The directing cosines of a vector are called the cosines formed by a vector with coordinate axes of Ox and Oy. Coordinates of a single vector coincide with its directing cosines. To find coordinates of a vector not equal to unit, it is necessary to increase the directing cosines by its length.

Author: «MirrorInfo» Dream Team