The essentials of descriptive geometry . The shortest distance from a point to a plane isthe perpendicular distance from the point to the plane. If theperpendicular be drawn from the point to the given plane andthe point where the perpendicular pierces the plane be found,the distance between these two points will be the requiredshortest distance. Construction. In Fig. 55 the problem is to find the altitudeof the pyramid whose base hes in the plane T and whose apex isat 0. From 0 draw a hne perpendicular to the plane T. (SeeObservation/, Article 23.) Find where this Hne pierces plane Tby Articl
The essentials of descriptive geometry . The shortest distance from a point to a plane isthe perpendicular distance from the point to the plane. If theperpendicular be drawn from the point to the given plane andthe point where the perpendicular pierces the plane be found,the distance between these two points will be the requiredshortest distance. Construction. In Fig. 55 the problem is to find the altitudeof the pyramid whose base hes in the plane T and whose apex isat 0. From 0 draw a hne perpendicular to the plane T. (SeeObservation/, Article 23.) Find where this Hne pierces plane Tby Article 56. p is the plan and p the elevation of this piercingpoint. Then op is the plan and op is the elevation of theshortest distance from the apex of the pyramid to the plane ofits base. Its true length may be found by Article 38, as shownat o^p. It should be noted that P, which is the foot of the perpendicu- PROBLEMS ON POINTS, LINES, AND PLANES 67 lar drawn from O to the plane T, is the projection of the pointO on the oblique plane Fig. 59. Corollary. Given a plane, the plan and elevation of theprojection of a point upon this plane, and its distance from theplane to find the plan and elevation of the point. Discussion. If a perpendicular be erected to the given planethrough this given projection of the point, the required point 68 ESSENTIALS OF DESCRIPTIVE GEOMETRY will lie in this line. To find its location a point on the perpen-dicular must be found at the given distance from the plane. Construction. Let the construction be made in accordancewith the suggestions in the discussion. 60. Special Cases, i. Find the shortest distance from a pointto a plane when the plane is parallel to the G. L. 2. Given two parallel planes to find the shortest distancebetween them. PROBLEMS 129. The plane S = o; —45; +60. The plane T = +3; —300; + the true length of that portion of the intersection of planes S and Twhich lies between H and V. 130. The plane S = 00 ; —3; —
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