I. Projection of a Point on a Line

A. TYPES OF PROJECTION

In plane geometry we know how to find the angle between two points, a point and a line, two lines. Also we learned how to find the distance between the points and lines. To find the same things in space geometry we need to use the projection. There are two types of projections; central projection and orthogonal projection.

Central Projection

If A is a fixed point not in plane α and P is any other point, the projection of P on α is point P' where line AP intersects α.

The central projection of a geometric figure upon a plane consists of the projections of all points of the figure. For example, the central projection of ΔABC from point P upon plane β is ΔA'B'C' (Figure 2.1).

The central projection of an object may not have the same shape as the object. For example, a central projection of a circle may be an ellipse. Projective geometry is a study which deals with the properties of figures which are not changed by projection.

Orthogonal Projection

A figure is projected orthogonally when the projecting lines are perpendicular to the plane of the projection. Since the projecting lines are perpendicular to the plane of the projection, they are parallel to each other.

In this book, we will use only orthogonal projection. Therefore, unless otherwise stated, “projection” will mean “orthogonal projection”.

A. Projection on a Line

i. Projection of a Point on a Line

Definition: (projection)

The projection of a given point on a line is the point which is the foot of the perpendicular drawn from the point to line.

In Figure 2.2, A' is the projection of A on line l and AA' is the distance from point A to line l.

If point A is on line l then its projection will be itself and the distance will be zero.

We can show the projection of point A on line l shortly as Proj _l A.

In this case Proj _l A = A'

Дата добавления: 2015-07-20