Projection

plane line figures lines

PROJECTION. If from a fixed point S in space lines or rays be drawn to different points A,B,C, . . . in space, and if these rays are cut by a plane in points A',B',C', . . . the latter are called the projections of the given points on the plane. Instead of the plane another surface may be taken, and then the points are projected to that surface instead of to a plane. In this manner any figure, plane or in space of three dimensions, may be projected to any surface from any point which is called the centre of pro-jection. If the figure projected is in three dimensions then this projection is the same as that used in what is generally known as perspective.

In modern mathematics the word projection is often taken with a slightly different meaning, supposing that plane figures are projected into plane figures, but three-dimensional ones into three-dimensional figures. Projec-tion in this sense, when treated by coordinate geometry, leads in its algebraical aspect to the theory of linear substi-tution and hence to the theory of invariants and co-variants.

In this article projection will be treated from a purely geometrical point of view.

We shall first and principally treat of the projection of plane figures into plane figures, and consider a number of special cases due to special positions of the two planes or of the centre of projection. W e shall next consider the representation of figures of three dimensions by plane figures (orthographic projections, drawing in plan and elevation, &c.), then treat of perspective in its ordinary sense, and speak shortly of projections to curved surfaces.

References like (G. § 87) relate to section II. of the article GEOMETRY, V01. X. pp. 388 sq.

§ 1. PnosEcTioN or PI.A:s7E FlGURES. - Let 115 5111)1)05C we have ill space two planes Ir aDd 7V. lu the plane 7r a figure is given having known properties ; then we hare the problem to find its projection from some centre S to the plane 71", and to deduce from the known properties of the given fig,nre the properties of the new one.

If a point A is given in the plane r we have to join it to the centre S and find the point A' where this ray SA cuts the plane ir'; it is the projection of A. On the other hand if A' is ,s,,iven the plane 7r; then A will be its projection in 7r. Hence our figure in. i8 the projection (g. another in 7r, then conversely the latter is also the projection of the former.

A point and its projection are therefore also called corresponding points, and similarly we speak of corresponding lines and curves,&e.

We at once get the following properties : - 7'he projection ■?t' a point is a point, and one point only.

111c projection of a lin, (straiyht ne) is a line ; for all points Ill a line are projecte 1 I.y rays which lie in the plane determined by S and the line, and this plane cuts the plane 7r in a line which is the projection of the given line.

//apoint /ies in a line its projection lies in the projection, of the line.

:The proj,....tiotr of the line joining two points A, B is the line which, yins the projections A', II' of the points A, For the projecting plane of the line All contains the rays SA, SI3 which project the points A, B.

The pr,iject ion of the point of intersection. of two lines a, b is the point qf intersection of the projections a', b' of those lines.

Sim ihrly we get - The projection of a Torre will be a curve.

The projections of the points of intersection, of two curves are the points of interst.,...tion of the po.ljections of the given, corms.

If a line cuts a curve in n points, then the projection of the line cuts the projection of the curve in n points. Or The order of a e?trre 2'010(12.0S unaltered by projection.

The projeetion of a tangent to a curve is a tangent to the projection of the curve. For the tangent is a line which has two coincident points in common with a curve.

The number of tangents that can be drawn from a point to a curve remains unaltered by projection. 01.

The class of a. curve 2'02220i21.3 unaltered by projection.

Example. - The projection of a circle is a curve of the second order and second class.

Two figures of which one is a projection of the other ob-tained in tho manner described inay be moved out of the position in which athey are obtained. They are then still said to Ile one the projection of the other, or to be projective or homographic. But when they are in the position originally considered they are said to 1.e in p-rspective position, or (shorter) to lw perspective.

All the properties stated in §§ 1, 2 hold for figures which are projective, whether they are perspective or not. There are others which hold only for projective figures when they are in perspective position, which we shall now consider.

If two planes 7r and 7r' are perspective, then their line of inter-sectim is called the axis of projection. Any point in this line coincides with its projection. Hence ..-1// points in, the axls are their 011! IL projections. Hence also Ecrry line meets its projection. 012. the eo..is.

The property that the lines joining corresponding points all pass through a common point, that any pair of corresponding points and the centre are in a line, is also .expressed by saying that the figures are co-linear ; and the faet that both figures have a.line, the axis, in common on which corresponding lines meet is expressed by saying that the figures are co-axal.

The connexion between these properties has to be investigated.

For this mirpose NYC consider in the plane ir a triangle Al3C, and let the lines IIC, CA, AB be denoted by a, b, c. The projection will conskt of three points A', C' and three lines a', b', e'. These have such a position that the lines AA', BB', CC' meet in a point, viz.. at S, and the points of intersection of a and a', b and b', c and lie on the axis (by § 2). The two triangles therefore are said to be both co-linear and co-axal. Of these properties either is a consequence of the other, will now he proved.

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