Problem I. Given the projections of three straight lines which meet in a point and are perpendicular to one another, on a plane oblique to each of them, to determine their inclinations to that Example I. To draw the projection of a cube of one inch side when ELEMENTS OF DESCRIPTIVE GEOMETRY. CHAPTER I. THEOREMS. THE STRAIGHT LINE AND PLANE. DEFINITION 1. A figure which has length, breadth, and thickness, is called a solid. DEF. 2. A surface is the boundary of a solid and has length and breadth only. DEF. 3. A plane is a surface such that if any two points be taken in it, the straight line passing through them lies wholly in that surface. The plane is said to contain the line. THEOREM I. Two straight lines which cut one another are in one plane. A E B D Let the two straight lines AB, CD, intersect in the point E, then AB and CD are contained by one plane. Proof. Let any plane which contains AB be made to revolve about that line as an axis, there is one position in which it will contain the point D; but it also contains the point E; therefore it contains the whole line CD, (Def. 3). Therefore AB and CD are in one plane. ...... Corollary. The line joining any two points B and D, one on each line, will be in the plane containing AB, CD,...... (Def. 3). Therefore three straight lines which meet one another, not in the same point, are in one plane. THEOREM II. If two planes cut one another, their common section is a straight line. Let M and N be the two intersecting planes, their common section is the straight line AB. Proof. Let A and B be two points common to both planes M and N. ...... Since the points A and B are in the plane M, the straight line joining them lies wholly in that plane, (Def. 3). Similarly, since A and B are in the plane N, the straight line AB lies wholly in N. Then the straight line AB is the common section of M and N. DEF. 4. The inclination to one another of two lines which do not meet is the angle contained by two intersecting lines parallel to them, each to each. DEF. 5. A straight line is perpendicular to a plane when it is at right angles to every line meeting it in that plane. The foot of the perpendicular is the point in which it meets the plane, and the line is called the normal to the plane at that point. DEF. 6. The angle between two intersecting planes is called a dihedral angle, and is measured by the angle between two straight lines drawn from any point of their common section, at right angles to it, one in each plane. DEF. 7. When the angle between two planes is a right angle, the planes are said to be perpendicular to one another. THEOREM III. If a straight line be perpendicular to each of two straight lines at their point of intersection, it shall also be perpendicular to their plane. A D B E Let AD be perpendicular to DB and DC at their point of intersection D; it is required to prove that AD is perpendicular to the plane BDC. = Let AD be produced to E, so that DE - DA, and E joined with B, C and F; DF being any line whatever in the plane BDC, and F on the straight line BC. Proof. In the two triangles ADB, EDB, AD=DE by construction, BD is common to both triangles, and the angle ADB=BDE, since each is a right angle. Therefore AB= BE. It may be proved in a similar way that AC=CE. Therefore the two triangles ABC, EBC have the three sides of the one respectively equal to the three sides of the other, and are consequently equal in every respect. Hence if the triangle EBC were turned about BC till the planes of the two triangles coincided, E would coincide with A and EF with AF. Therefore AF = EF. Then in the triangles ADF, EDF the three sides of the one are respectively equal to the three sides of the other. Therefore the angle ADF = EDF. That is ADF and EDF are right angles. Therefore AD is perpendicular to DF. In a similar way it may be shown that AD is perpendicular to every line passing through D in the plane BDC, and is therefore perpendicular to the plane. Cor. 1. It follows from Def. 4 that if AD is perpendicular to any two lines in the plane it is perpendicular to every line in it. Cor. 2. Any number of straight lines which are drawn at right angles to the same straight line from the same point of it, are all in the plane which is perpendicular to the line at that point. Hence if one of the arms of a right angle be made to revolve about the other as an axis, it will describe a plane normal to that axis. Cor. 3. Through any given point, either within or without a plane, only one normal can be drawn to the plane. For if it were possible to draw more than one, each of them would be perpendicular to a straight line in the same plane with them, which is impossible. |