19. Through a given point, in the plane of the primitive, to describe the projection of a great circle, having a given inclination to the primitive. THIRD BOOK. PERPENDICULAR PROJECTION. DEFINITIONS. 1. In Perpendicular Projection, the position of the point of sight is at an indefinitely great distance in the axis of the primitive. 2. The perpendicular projection of any point is called the seat of that point. · 3. Two planes at right angles are sometimes assumed for planes of projection, and they are then called co-ordinate planes. 4. When the two co-ordinate planes are a horizontal and a vertical plane, the projections on them are respectively called the horizontal and vertical projection. 5. In the case of an object of a regular figure, such as a house, the horizontal projection is called the plan ; and the vertical, the elevation. When the projection is that of a section, it is called a section. Cor. 1.-A projecting line in perpendicular projection is a line drawn from the original point perpendicular to the primitive (Def. 1). Cor. 2.-If the seats of a point on any two planes, not parallel, be given, the position of the point in space may be found; for it will be the point of intersection of the two projecting lines drawn from the two seats. Cor. 3.-If the projections, on two planes, of all the remarkable points and lines of an object, be given, the figure and dimensions of the object may be determined. This method of projection is employed in the construction of plans and sections of houses, the plans of philosophical apparatus, and of various objects in engineering and carpentry. A LA PROPOSITION I. The projection of a line inclined to the primitive is less than the line in the ratio of radius to the cosine of its inclination. · Let A'B' be a line inclined to the primitive MR, and AB its projection, and A'C parallel to AB, then A'B': AB = Radius : cosine B'A'C. For, the projecting lines A’A, and B’B, being perpendicular to the primitive, are parallel; and the angles at A and B are right angles, and A'C is pa- L rallel to AB; therefore AC is a rectangle, and A'C = AB; and the inclination of A'B' to A'C is the same as its inclination to AB or to the primitive. Also, A'B': A'C= Radius : cosine B’A'C; or A'B': AB = Radius : cosine of inclination. Cor. 1.-If a line parallel to the primitive join any two projecting lines, and if a line on the primitive parallel to it join the same lines, these lines, with the inter cepted portions of the projecting lines, form a rectangle. For AC is a rectangle. COR. 2.-The projection of a line parallel to the primi tive, is equal to the original. For the projection of A'C is AB, and they are equal. CoR. 3.-The projection of a line perpendicular to the primitive, is a point. PROPOSITION II. The projection of a plane figure parallel to the primitive, is a figure similar and equal to the original. Let ABCDE be the projection of M_ A'B'C'D'E', which is parallel to the primitive, these figures are similar and equal. For the sides AB, BC, &c. are equal to A'B', B'C', &c. (Pr. III. 1, Cor. 1). Also, if any plane perpen- L dicular to the primitive cut the two figures in lines B'E', BE, these lines are corresponding dimensions of the two figures, and they are equal. Also, any two lines in one of the figures, as A'B', B’E', have the same inclination as the corresponding lines of the other figure AB, BE. . · CoR.-The intersection of the primitive, with a plane per pendicular to it, is the projection of this plane's intersection with any other plane or surface or solid figure. PROPOSITION III. If a plane figure be inclined to the primitive, any of its dimensions parallel to the line of common section of its plane with the primitive, is equal to its projection ; but any of its dimensions perpendicular to this line exceeds its projection in the ratio of radius to the cosine of obliquity. · Let ON be the intersection of a plane figure with the primitive MR; A'B' a section of it by a projecting plane paral- M lel to ON; and C'D' a section A of it by a projecting plane perpendicular to ON, cutting ON | in 0. Draw projecting lines from A', B', C', and D'; and let AB be the projection of A'B'; . CD of C'D'; and OC of OC'. L Then, since ON is perpendicular to the plane ODD', it is so to the lines OD, OD'; and therefore, the inclination of the plane of the figure with the primitive is measured by the angle DOD'. Now (Pr. III. 1) OD': OD = Radius: eosine DOD'; but OD': OD = ('D' : CD. Therefore C'D': CD = Radius : cosine DOD'. - Again, since A'B' is parallel to AB, AB' is a rectangle, and AB = A'B'. Cor. 1.- The projection of a plane figure inclined to the · primitive, is less than the original in the ratio of the radius to the cosine of obliquity. For every line perpendicular to the line of common section exceeds its projection in this ratio ; and the lines parallel to the line of common section are equal to their projections; hence the whole figure is greater than its projection in this ratio. AR Cor. 2. The projection of a circle inclined to the primi. , tive is an ellipse, unless it be a subcontrary section, in C, which case it is a circle. For, when its projection is a subcontrary section of the cylindrical projecting surface, it is a circle (Conic Sections, IV. 2). Again, when the circle is inclined, its diameter, which is parallel to the line of common section of its plane with the primitive, is projected into an equal line; and all the dimensions perpendicular to this line are projected into less lines in the ratio of radius to the cosine of obliquity. Hence the projection is an ellipse, of which the transverse axis is equal to the diameter of the circle, and its conjugate less in the preceding ratio. Cor. 3.--The projection of a sphere is a circle of equal diameter. FOURTH BOOK. DEFINITIONS. 1. The theory of Linear Perspective treats of the method of projecting objects on a vertical plane from some given point of sight. 2. The plane of projection is also called the perspective plane, or the plane of the picture. 3. The point of sight is also called the point of view. 4. The centre of the perspective plane is also called the centre of the picture. 5. The distance of the point of view from the primitive, is called the distance of the picture. 6. A vertical plane passing through the axis of the primitive, is called the vertical plane ; and a horizontal plane passing through it, is called the horizontal plane. 7. The intersection of the primitive with the horizontal plane, is called the horizontal line ; and its intersection with the vertical plane is called the vertical line. 8. The intersection of the primitive with the ground plane, or that of the sensible horizon, is called the ground line , · 9. The intersection of the primitive with a line from the point of view parallel to any original line, is called the vanishing point of that line. 10. Two points on the horizontal line, whose distances from the centre are equal to the distance of the picture, are called points of distance. PROPOSITION I. If the distance between the centre and the seat of any point be cut in the ratio of the distance of the picture to the distance of the point from the primitive, the point of section will be the perspective of the point. Let MR be the primitive, C its centre, S the point of sight, P' the given point, and N its seat. Then, if the point P in NC be taken so that CP: PN = SC:P'N, P is the perspective of 5 For SC and NP' are parallel, and therefore in the same plane, and they are in the same plane with CN. Also, the angles C and N are equal, being right angles, and SC : CP = P'N:NP; and hence the triangles SCP and P'NP are similar. Therefore the angles at P are equal, and therefore SP, PP', are in one straight line; and P is therefore the perspective of P'. PROPOSITION II. The sum of the distance of the picture, and the distance of any point from it, is to the distance of the picture, as the distance of that point from the vertical plane to the distance of its projection from the vertical line ; and also as the distance of that point from the horizontal plane to the distance of its projection from the horizontal line. Let MR be the primitive, P' the given point, P its projection, and S the point of sight, and C the centre. Through P' let a plane P'o pass parallel to MR, cutting the horizontal and vertical planes in the lines B'C' and C'A', and in that plane draw P'A' and P'B' parallel to the opposite sides of the figure P'C'; join S and the points B', P', A' and let these lines cut the primitive in B, P, A. |