The rules of solid geometry apply to 3-D objects and to the position(s) of other geometrical elements in 3-D space.
Sphere
A sphere, also known as a ball, is a closed, uniformly curving, three-dimensional surface; all points on its surface are the same distance from an interior point, the center.
Ellipsoid
An ellipsoid is the solid figure formed by rotating an ellipse about one of its two axes. An ellipse rotated about its major axis looks rather like a football. An ellipse rotated about its minor axis is used to approximate the size and shape of the Earth.
Polyhedron
A polyhedron is a solid figure bounded by plane surfaces—usually more than six. Tetrahedron
A tetrahedron is a regular solid whose sides consist of four equilateral triangles. It is a figure formed by the minimum number of plane sides.
Pyramid
A pyramid is a solid whose base is a polygon and whose triangular sides meet at a common point. Most pyramids are five-sided, having a square base and four triangular sides.
Cube
A cube is a polyhedron having six identically square faces (sides).
Equation of a Plane in Space
A plane in space is described by a first-order (containing only powers of 1 for each variable) equation using variables X/Y/Z as
The letters a, b, and c are coefficients of the variables and d is a constant; all of them are real numbers. Note that the entire equation could be divided by d, giving equation 3.5 having only three independent coefficients. That makes sense, because it takes three points in space to determine a plane.
Equation of a Sphere in Space
A sphere in space is described by a second-order (contains powers of 2 on one or more variables) equation using variables X/Y/Z as
The letters a, b, and c are the XIYIZ coordinates of the center of the sphere, and R is the radius of the sphere. If a, b, and c are all zero, the center of the sphere lies at the origin of the coordinate system.
Equation of an Ellipsoid Centered on the Origin
A two-dimensional ellipse rotated about its major axis forms a football-shaped figure. A two-dimensional ellipse rotated about its minor axis is used to approximate the Earth’s size and shape. The North Pole and South Pole lie on the Earth’s spin axis, which is the minor axis of the ellipsoid.
The letter a is used as the semimajor axis, and b is the semiminor axis of the ellipse. The equatorial plane goes through the origin and is 90° from each pole. The equator forms a circle having a as its radius. Each meridian section is perpendicular to the equator and forms an ellipse defined by the letters a and b.
Conic Sections
A cone is a triangular-shaped solid whose base is a closed curve. A right circular cone is one whose base is a circle that is perpendicular to the cone axis. Conic sections are two-dimensional shapes obtained by intersecting a cone with a plane at different orientations.
1. A circle is formed by the intersection of a cone with a plane perpendicular to the axis of the cone. If the intersection occurs at the vertex, the circle is reduced to a point.
2. An ellipse is formed by the intersection of a cone with a plane that is not perpendicular to the cone axis. The intersection is a closed figure.
3. A parabola is formed by the intersection of a cone with a plane that is parallel with the opposite side of the cone. The intersection is not a closed figure.
4. A hyperbola is formed by the intersection of a cone with a plane that is parallel with the axis of the cone. The intersection is not a closed figure.
Conic sections can all be derived from the general second-degree polynomial equation by the appropriate selection of coefficients, A, B, C, D, E, and F. It is to be understood that the XIY coordinate system used in equation 3.8 lies in the plane intersecting the cone.
Vectors
A vector is a directed line segment in space. In terms of a right-handed coordinate system, a vector is composed of signed components in each of the i/j/k directions. The length of a vector is called its magnitude and is computed as the square root of the sum of the components squared (a three-dimensional hypotenuse). A unit vector has a length of 1.0 and is obtained by dividing each component by the vector magnitude. The resulting rectangular components of a unit vector are its direction cosines. The underlying i/j/k orientation of a vector can be rotated to any X/Y/Z coordinate system without changing the length or statistical qualities of the vector. The direction cosines do change.
Vectors can be added and subtracted component by component (subtraction is the same as addition given that the sign is changed for each component of the second vector).
A vector can be multiplied by a scalar that has the effect of changing only the magnitude of the vector. The direction cosines remain the same.
Vector multiplication takes two forms: a dot (inner) product of two vectors is a number (scalar) that can be used to find the angle between two vectors in a plane and the cross product of two vectors that is a third vector perpendicular to the plane common to the first two.
