We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.

Transcription

1 Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the Euclidean geometry of the plane holds in every plane in space. This is called the principle of homogeneity. In many cases, properties of objects in space are simple extensions of properties in the plane, but sometimes there are unexpected issues. Lines are perpendicular in space if they lie in the same plane and are perpendicular in that plane. Also, a line is perpendicular to a plane at a point in that plane if it is perpendicular to every line in the plane that passes through that point. Finally, the plane P is perpendicular to Q if P contains some line l that is perpendicular to Q. Theorem Given a point A not on a plane P, there exists a unique line l through A perpendicular to P. // Theorem [The Fencepost Property] If a line l is perpendicular to two different lines m and n in the same plane, then it is perpendicular to every line in the plane through that point, and is perpendicular to the plane there as well. //

2 Corollary Perpendicularity of planes is symmetric: if P Q then Q P. // Theorem If perpendicular planes P and Q intersect in a line l, then every line in P perpendicular to l is perpendicular to Q. // Corollary If planes P and Q intersect in a line l and are both perpendicular to a third plane R, then l is perpendicular to R. // Theorem The set of points equidistant to two points A and B is the plane perpendicular to A B a its midpoint. // In the same way that the Plane Separation axiom asserts that a line separates any plane in which it lies into three disjoint convex sets the line itself and two half planes we will need a similar Space Separation axiom to have the analogous property in the higher dimensional setting: [H-2] A plane P cuts space into three convex subsets: P itself and two disjoint half spaces Σ 1 and Σ 2. Further, given any points A Σ 1 and B Σ 2, the segment A B meets P. We can now define the analog of an angle between two lines: if l is the line of intersection of two

3 planes P and Q, then a dihedral angle is the union of l with one of the half planes H 1 determined by l in P and one of the half planes H 2 determined by l in Q; H 1 and H 2 are the sides of the dihedral angle, which we denote (H 1, H 2 ), and l is its edge. Theorem If P and Q are two planes perpendicular to the edge l of a dihedral angle (H 1, H 2 ), then the angles formed by the intersection of P and Q with (H 1, H 2 ) are congruent. // Lines are parallel in space if they lie in the same plane and are parallel in that plane (i.e., they do not meet); planes are parallel if they do not meet; and a line and plane are parallel if they do not meet. Lemma If line l is parallel to plane P, then it is parallel to every line in P that is coplanar with l. // Theorem If line l in plane P is parallel to line m in plane Q and P Q, then either P Q or they intersect in a line n that is parallel to both l and m. // Corollary Given three distinct lines l, m, n, if l m and m n, then l n. //

4 Polyhedra A (convex) polyhedron is a finite union of convex polygons with their interiors, each of which is a face of the polyhedron, having the property that all points of the polyhedron lie in the same half space determined by any face. The edges of a face and the vertices of an edge are also called edges and vertices of the polyhedron. The union of a polyhedron with its interior (the intersection of the half spaces determined by its faces that contain the polyhedron) is called a polytope. A tetrahedron is a polyhedron with four triangular faces. A box (or hexahedron) is a polyhedron with six quadrilateral faces. A box whose faces are all parallelograms is a parallelepiped; if the faces are all rectangles, we call it a rectangular box, and if all faces are squares, it is a cube. A prism is a polyhedron with two faces (called bases, the bottom and top) in parallel planes; the other faces are called lateral sides and must lie in planes parallel to a fixed line. We identify a prism by the geometry of its bases, e.g., a pentagonal prism has pentagonal bases.

5 Theorem The lateral faces of a prism are (closed) parallelograms, and its top and bottom are congruent (closed) polygons. // Corollary A prism is a parallelepiped if and only if its bases are parallelograms. // A pyramid is a polyhedron with a distinguished face called its base; the remaining faces, its lateral sides meet in a common vertex, called the apex. Theorem The lateral faces of a prism are (closed) triangles. // Theorem The sum of the measures of the angles on the faces of a polyhedron which are made at their common vertex have angle sum less than 360. // Theorem The measures of the angles on the faces of a tetrahedron which are made at their common vertex satisfy the triangle inequality: the sum of any two is larger than the third. //

6 The Platonic solids are polyhedra with congruent regular (closed) polygons for faces. A classical result in solid geometry (one of the oldest known) is Theorem There are precisely five types of Platonic solid, the regular tetrahedron (with 4 equilateral triangles as faces), the cube (with 6 squares as faces), the regular octahedron (with 8 equilateral triangles as faces), the regular dodecahedron (with 12 regular pentagons as faces), and the regular icosahedron (with 20 equilateral triangles as faces). // A more modern but no less classic result is generally attributed to Euler. Theorem [Euler Characteristic for Polyhedra] Let a polyhedron have F faces, E edges and V vertices. Then its Euler characteristic, the quantity F E + V, equals 2. // Let γ be a curve in the plane P and l any line not parallel to P. Then the union of all the lines parallel to l and passing through any point of γ is called a cylindrical surface with base curve γ ; any one of the lines that makes up the cylindrical surface is called a line generator for the surface. When γ is a circle, the cylindrical surface is a circular cylinder, or simply a cylinder, and the

7 line through the center of γ parallel to the line generators is called the axis of the cylinder. If the line generators are perpendicular to the plane P containing γ, we call the cylinder a right circular cylinder. Let γ be a curve in the plane P and A any point not on P. Then the union of all the lines passing through A and any point of γ is called a conical surface with base curve γ and vertex (or apex) A, and any one of the lines that makes up the conical surface is a line generator for the surface. When γ is a circle, the conical surface is a circular cone, or simply a cone, and the line through the center of γ passing through A is called the axis of the cone. The two portions of the cone on either side of the vertex are called the nappes of the cone. If the line generators are perpendicular to the plane P containing γ, we call the cone a right circular cone. The set of points a fixed (positive) distance r from a fixed point O is the sphere with radius r and center O. The points whose distance to O is less than r is the interior of the sphere and the points a distance greater than r from O is its exterior. The union of the sphere with its interior points is called a ball.

8 Theorem If a plane meets a sphere, it does so in a single point, in which the plane is said to be tangent to the sphere at that point, or in a circle, in which case the plane is secant to the sphere. The center of such a circle of secancy lies on a line perpendicular to the plane and passing through the center of the sphere. // Any circle which is the intersection of a plane with a sphere (and, therefore, by the last theorem, whose center lies on a radius of the sphere) is called a small circle on the sphere if the plane does not also pass through the center of the sphere, and a great circle if it does. The next theorem is actually a theorem of absolute geometry: Theorem If two spheres, Σ 1 with center O 1 and Σ 2 with center O 2, meet, their intersection is either a point, in which case the plane tangent to Σ 1 at this point is also tangent to Σ 2 there, or a circle. In either case, the plane containing the circle of intersection is perpendicular to the line O s r 1 O 2 of centers. // Theorem Any four noncoplanar points determine a sphere. //

9 Volume Recall that we gave a set of axioms for volume in space. These now come into play. We attempt to determine the measures of the standard bounded convex solid regions in space. Lemma The volume of any planar cross section (intersection with a plane) of a finite union of bounded convex solid regions is always 0. Theorem Let Π be a prism with base B and altitude h (the perpendicular distance between the planes that contain the base and top of Π). If B has area B, then Vol(Π) = Bh. // Theorem Let Κ be a circular cylinder with base circle C, having radius r, and altitude h. Then Vol(Κ) = πr 2 h. // Theorem Let Λ be a triangular pyramid (i.e., a tetrahedron) whose base triangle has area B, and whose altitude is h (the perpendicular distance between the parallel planes containing the base and vertex of the pyramid). Then Vol(Λ ) = 1 Bh. // 3