Modern geometry . iangle^(v) right-angled triangle,(vi) parallelogram,(vii) rectangle,(viii) rhombus,(ix) trapezium,(x) circle,(xi) a set of equivalent triangles, on the same base and on the same side of it,(xii) a set of triangles with the same base and equal vertical angles. Ex. 514. If the original plane is covered with squared paper, what isthe corresponding pattern on the plane of projection ? Ex. 515. If a, triangle is projected orthogonally, the centroid of thetriangle projects into the centroid of the projection. The Ellipse. The most interesting application of the method of orthogonal
Modern geometry . iangle^(v) right-angled triangle,(vi) parallelogram,(vii) rectangle,(viii) rhombus,(ix) trapezium,(x) circle,(xi) a set of equivalent triangles, on the same base and on the same side of it,(xii) a set of triangles with the same base and equal vertical angles. Ex. 514. If the original plane is covered with squared paper, what isthe corresponding pattern on the plane of projection ? Ex. 515. If a, triangle is projected orthogonally, the centroid of thetriangle projects into the centroid of the projection. The Ellipse. The most interesting application of the method of orthogonalprojection is that derived from the circle. The circle projects into an oval curve called an ellipse; it isflattened or foreshortened along the lines of steepest slope, whilethe dimensions parallel to the plane of projection are unaltered. If we define the ellipse, for present purposes, as the curvewhose equation is a* b~ it is easy to prove that the ellipse is the projection of a circle. 120 OBTHOGONAL PROJECTION. fig. 69. Let the circle (centre O) be referred to rectangular axes OX,OY; OX being || to the plane of projection. The coordinates of a point p on the © are On, On — x,pn = Y, radius = a. Then oi? + y^ = a\ The projections of OX, OY are the perpendicular lines CA, CB;these shall be the axes for the ellipse. The coordinates of the point P on the ellipse are ON, ON = Ow = PH= y = Y cos 6, a? + r a? cos^e = 1. a? a^coae But CB, the projection of OY, = a cos 0. Let CB = the coordinates of P satisfy the equation ^^ 4. ^ _ 1 The locus of P is therefore an ellipse whose semiaxes areCA (a) and CB (6). The angle properties of the circle do not admit of transferenceto the ellipse. But there are many important properties that ORTHOGONAL PROJECTION - 121 may be transferred, and the chief of these are given in thefollowing exercises. Ex. 516. Prove the following properties of the ellipse, byfirst proving the allied property of the circle, and th
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