Mathematical Handbook for Scientists and Engineers

3.1-1. Introduction (see also Sec.2.1-1). Chapter 3 deals with the analytic geometry of three-dimensional Euclidean space, corresponding to classical (Euclidean) solid geometry. The elements of the description are points labeled by sets of real numbers (coordinates) or represented by position vectors.

3.1-2. Cartesian Coordinate Systems (see also Sec.2.1-2). A cartesian reference system or cartesian coordinate system associates a unique set of three real numbers (cartesian coordinates) x, y, z with every point P ≡ (x, y, z) in the finite portion of space by reference to three noncoplanar directed straight lines (cartesian coordinate axes) OX, OY, OZ intersecting at the origin 0 (Fig.3.1-1a).

FIG.3.1-la. (Oblique) right-handed cartesian coordinate system. The points marked “1” on the x, y, z axes define the coordinate scales used.

Given a (in general oblique) system of coordinate axes OX, OY, OZ (Fig.3.1-la), the (unique) plane through the point P and parallel to the yz plane YOZ intersects the x axis OX at P′. Similarly, the planes through P respectively parallel to the xz plane XOZ and parallel to the xy plane XOY intersect the y axis OY at P″ and the z axis OZ at P′″. Each directed distance OP′ = x, OP″ = y, and OP′″ = z is given the positive sign if its direction coincides with the respective positive direction of the x axis, the y axis, and the z axis, x, y, and z are the cartesian coordinates of the point P ≡ (x, y, z) with respect to the reference system defined by the given coordinate axes and the given (not necessarily equal) scales used to measure x, y, and z.

3.1-3. Right-handed System of Axes (Fig.3.1-la). The ordered set of three directed straight lines (cartesian axes) OX, OY, OZ is right-handed if and only if the rotation needed to turn the x axis OX into the y-axis direction OY through an angle deg would propel a right-handed screw toward the side (positive side) of the xy plane XOY associated with the positive z axis OZ.

3.1-4. Right-handed Rectangular Cartesian Coordinate System. In a rectangular cartesian coordinate system, the coordinate axes are mutually perpendicular (Fig.3.1-1b). The coordinates x, y, z of a point

FIG. 3.1-1b. Right-handed rectangular cartesian coordinate system, cylindrical coordinate system, and spherical coordinate system.

P are equal to the directed distances, measured on suitable scales, between the origin and the yz plane, the xz plane, and the xy plane, respectively.

Throughout this handbook all coordinate values x, y, z refer to right-handed rectangular cartesian coordinate systems,and equal scale units of unit length are used for measuring x, y and z, unless otherwise specified.

3.1-5. Position Vectors. Each point P ≡ (x, y, z) ≡ (r) may be represented uniquely by the position vector

which may be represented geometrically by the translation. The base vectors i, j, k are unit vectors directed along the x, y, z axes, respectively (see also . Secs5.1-1 and Secs5.2-2), of the right-handed rectangular cartesian coordinate system used.

3.1-6. Cylindrical and Spherical Coordinate Systems . Figure 3.1-1b also illustrates cylindrical coordinates r′, φ, z and spherical coordinates r, ϑ, φ (radius vector, colatitude, and longitude) defined so that

For a further discussion of these and other noncartesian coordinate systems, refer to Chap. 6.

3.1-7. Basic Relations in Terms of Rectangular Cartesian Coordinates and Position Vectors . In terms of rectangular cartesian coordinates x, y, z and position vectors r ≡ (x, y, z), the following relations hold:

1. The distance d between the points P₁ and P₂ specified by is

2. The angle γ between the straight lines defined by two directed line segments and is given by

where the coordinates or position vectors of the points P₁ P₂, P₃, P₄ are denoted by the corresponding subscripts.

If the straight lines P₁P₂ and P₃P₄ do not intersect,γ is defined as the angle between two intersecting straight lines parallel to P₁P₂ and P₃P₄, respectively.

3. The coordinates x, y, z and the position vector r ≡(x, y, z) of thepoint P dividing the directed line segmentin the ratio

Specifically, the coordinates and the position vector of the mid-point of the line segment P₁P₂ are given by

3.1-8. Direction Cosines and Direction Numbers. (a) The direction cosines cos α_x, cos α_y, cos α_z of a directed line segmentare the cosines of the angles α_x, α_y, α_z betweenand the positive x, y, and z axis, respectively:

The direction cosines of the directed line segment are — cos α_yi — cos α_z.

(b) Any set of three numbers α_x, α_y, α_z proportional (with the same proportionality constant) to the direction cosines cos α_x, cos α_v, cos α_zare called direction numbers of the line in question, and

where the sign of the square root is taken to be the same in all three equations; its choice fixes the positive direction of the line. The components of any vector a (Sec.5.2-2) having the same direction as either or are direction numbers of . (EXAMPLE: x₂ – x₁, y₂ – y₁,z₂ – z₁.) The direction cosines of are the components of a unit vector directed along ( Sec.5.2-5).

(c) The angle γ between two line segments having the respective direction cosines cos a_x, cos a_y , cos a_z and cos a′_x1 cos _y1 cos a_z or the respective direction numbers a_x, a_y, a_z and a_x, a′_yi a′_z is given by (see also Secs.3.1-7 and Secs.3.4-1)

3.1-9. Projections. The respective projections of a line segment having the length (4) and the direction cosines (8) onto the x, y, and z axes are

The projection of on to the straight line having the direction cosines cos α′_x, cos α′_y, cos α′_z (Sec. 3.2-1b) is d cosα,where cos γ is given by (Eq.11).

The respective projections of onto the yz, xz, and xy planes are

The projection of onto a plane whose normal has the direction cosines cos α′_x,cos α′_y, cos α′_z (Sec.3.2-1b) is d sinγ,where cos γ is given by (Eq.11).

3.1-10. Vector Representation of Areas. A (finite) plane area in three-dimensional Euclidean space may be represented by a vector A of magnitude equal to the area A and directed along the positive normal (Sec.17.3-2) of its plane:

where N is a unit vector directed along the positive normal. The rectangular cartesian components of A are the projections of the area onto the coordinate planes. The area of the parallelogram formed by the vectors a and b may be represented by A = a × b (see also Sec.5.2-7).

The area A of a triangle with the vertices P₁, P₂, P₃ is given by

where the coordinates or position vectors of the points P₁, P₂, P₃ are denoted by the respective corresponding subscripts, and where p is the distance between the origin and the plane of the triangle (Sec.3.2-1b). A is positive if a right-handed screw turning in the direction P₁P₂P₃ is propelled in the direction of the positive normal of the triangle plane (Sec.3.2-1b).

3.1-11. Computation of Volumes (see also Sec.5.2-8). (a) The volume V of the tetrahedron with the vertices P₁, P₂, P₃, P₄ is

where the coordinates or position vectors of the points P₁, P₂, P₃, P₄ are denoted by the corresponding subscripts. The sign of V depends on the order of the vertices.

(b) The volume V of the parallelepiped formed by three vectors a, b, c is

3.1-12. Translation and Rotation of Rectangular Cartesian Coordinate Systems. (a) Translation of Coordinate Axes. Let x, y, z be the coordinates of any point P with respect to a right-handed rectangular cartesian reference system. Let ,, be the coordinates of the same point P with respect to a second right-handed rectangular cartesian reference system whose axes have the same directions as the corresponding axes of the x, y, z system, and whose origin has the coordinates x = x_o, y=y₀, z=z₀ in the x, y, z system. If equal scales are used to measure the coordinates in both systems, the coordinates ,, are related to the coordinates x, y, z by the transformation equations

(b) Rotation of Coordinate Axes. Given any point P ≡ (x, y_} z), let ,, be the coordinates of the same point with respect to a second right-handed rectangular cartesian reference system having the same origin 0 as the x, y, z system and oriented with respect to the latter so that

The axis has the direction cosines t₁₁, t₂₁, t₃₁.

The axis has the direction cosines t₁₂, t₂₂, t₃₂.

The axis has the direction cosines ₁₃,t₂₃, t₃₃.

Then in the ,,system

The x axis has the direction cosines t₁₁,t₁₂, t₁₃.

The y axis has the direction cosines t₂₁,t₂₂, t₂₃.

The z axis has the direction cosinest₃₁,t₃₂, t₃₃.

If equal scales are used to measure x, y, z, x, y, and z, the transformation equations relating the coordinates ,, to the coordinates x, y, z are

NOTE: The transformations (20) are orthogonal (Secs.13.3-2 and 14.4-5); each t_ik equals the cofactor of t_ki in the determinant

( Sec.1.5-2), and

(c) Simultaneous Translation and Rotation.If the origin of the x, , z system is not that of the x, y, z system but has the coordinates x=x₀, y= y₀,z=z₀ in the x, y, z system, the transformation equations become

where the direction cosines t_iksatisfy the relations (21) and (22).

The equations (23) relate the coordinates of a point in any two right-handed rectangular cartesian reference systems if the same scales are used for all coordinate measurements. More general types of coordinate transformations are discussed in Chap. 6.

(d) Alternative Interpretation of Coordinate Transformations. The transformation equations (23) [of which (19) and (20) are special cases] may also be interpreted as defining a new point P having the coordinates x, , z in the x, y, z system and obtained by successive translation and rotation of the original point P (see also Secs.14.1-3 and 14.5-1).

NOTE: The values of distances between points, angles between line segments, and thus, in fact, all relations between position vectors constituting Euclidean geometry are unaffected by (invariant with respect to) translations and rotations of coordinate systems (see also Secs. 12.1-5 and 14.1-4). Refer to Chap. 14 for transformation laws of vector components and base vectors.

3.1-13. Analytical Representation of Curves (see also Secs. 17.2-1 to 17.2-6). A continuous curve in three-dimensional Euclidean space is a set of points (x, y, z) ≡ (r) whose coordinates satisfy a system of parametric equations

where x(t), y(t), z(t) are continuous functions of the real parameter t throughout the closed interval [t₁ t₂] (parametric representation of a curve). Alternatively, a curve may be defined by an equivalent set of two equations

A curve can have more than one branch; branches may or may not be connected.

A simple curve (simple arc, simple segment) is a (portion of a) continuous curve consisting of a single branch without multiple points, so that the functions (24) are single-valued, and

is not satisfied by any pair of values r₁ ≠ r₂ of t in the closed interval [t₁ t₂]. A simple closed curve is a continuous curve without multiple points except for a common initial and terminal point; i.e., the only solutions of (Eq. 26) in the closed interval [t₁ t₂] are r₁ = t₁, r₂ = t₂. A regular arc is a continuous curve which can be represented in terms of some parameter t so that every point of the curve is a regular point where x(t), y(t), z(t) have unique continuous derivatives (Sec.4.5-1) not all equal to zero. A regular curve is a simple curve or a simple closed curve composed of a finite number of regular arcs. Such a curve is also called piece wise smooth.

NOTE: Analogous definitions apply to curves in two-dimensional spaces (Secs.2.1-9 and 7.2-1) and also to curves in spaces of dimensionality higher than three (Sec.17.4-2).

Secs.3.1-14. Representation of Surface (see also . 17.3-1 to 17.3-13). The coordinates of each point (x, y, z) ≡ (r) on a continuous surface in three-dimensional Euclidean space satisfy a set of parametric equations

for suitable ranges of the real parameters u, v; the functions (27) are continuous (parametric representation of a surface). A surface can also be defined by an equation

A surface can have more than one sheet; sheets may or may not be connected. The surfaces corresponding to the equations

are identical for any constant λ different from zero.

A simple surface (simple surface element) is a (portion of a) continuous surface consisting of a single sheet without multiple points (Sec.3.1-13). A simple closed surface is a continuous surface which consists of a single sheet and has no multiple points except for the points of one simple closed curve (Sec.3.1-13); a simple closed surface bounds a connected region of space (Sec.4.3-6b). Simple surfaces and simple closed surfaces are understood to be two-sided (one-sided surfaces such as the Moebius strip are excluded). A regular point of a surface (27) is a surface point where, for some choice of the parameters u, v, the functions (27) have continuous partial derivatives such that at least one of the determinants

is different from zero. A regular surface element is a simple surface element which has only regular points and is bounded by a regular closed curve (Sec.3.1-13). A regular surface is a two-sided simple surface or simple closed surface comprising a finite number of regular surface elements joined along regular arcs (edges) and/or points (vertices). Such a surface is also called piecewise smooth.

3.1-15. Special Types of Surfaces A ruled surface is a surface generated by the displacement of a straight line (generator, generatrix). A cylinder is a surface generated by the displacement of a straight line parallel to itself along a directing curve (directrix). In particular, equations of the form f(x,y) = 0, f(x, z) = 0, f(y, z) = 0 describe cylinders whose generators are perpendicular to the xy plane, the xz plane, and the yz plane, respectively. A cone is a surface generated by a straight line through a fixed point (vertex) and a point of a directing curve (directrix).A surface of revolution is the locus of the points of a curve rotated about a straight A surface of revolution is the locus of the points of a curve rotated about a straight line (axis of revolution). Thus, describes a surface of revolution about the z axis.

3.1-16. Surfaces and Curves, (a) The intersection of two surfaces

is the locus of the points (x, y, z) which satisfy both equations (31) (see also Sec.3.1-13). A curve of intersection can have more than one branch; or it may degenerate into a set of points where the two surfaces (31) touch.

Specifically, the curves

represent the intersections (if they exist) of the surface described by φ(x, y, z) =0 with the yz, xz, and xy planes, respectively.

(b) For any real number λ, the equation

corresponds to a surface passing through all points of the curve of intersection of the two surfaces (31) if that curve exists.

(c) Given two surfaces corresponding to the two equations (31), the

corresponds to the surface made up of all the points of both original surfaces, and no other points.

(d)The respective equations of the projections of a curve described by the equations (31) on the yz, xz, xy planes are obtained by eliminating x, y, z, respectively, from the equations (31).

(e)The coordinates x, y, z of the point(s) of intersection of a curve described by Eq. (31) with a surface described by Eq. (28) satisfy all three equations (28) and (31). An n^th-order curve intersects a plane (Sec.3.2-1) in n points, some of which may coincide and/ or imaginary.For Any real λ₁,λ₂,the equations

represent a curve passing through all these points of intersection (if they exist).

(f) A suitable one-parameter family of curves described by described by

generates a surface whose equation may be obtained by eliminating λ from the equations (34).

3.2. THE PLANE

3.2-1. Equation of a Plane. (a) Given a right-handed rectangular cartesian coordinate system, an equation linear in x, y, z, i.e., a relation of the general form

where A, B, and C must not all vanish simultaneously, represents a plane; conversely, every plane situated in the finite portion of space can be represented by a linear equation of the form (1).

A, B, C are direction numbers (Sec.3.1-8b) of the (positive or negative; see below) normal to the plane; the vector A ≡ (A, B, C) is directed along the normal. The special case D = 0 corresponds to a plane through the origin.

(b) The following special forms of the equation of the plane are of interest:

(c)When the equation of a plane is given in the general form (1), the quantities a, b, c, p, cos α_x cos α_y cos α_z defined above are related to the parameters A, B, C, D as follows:

where the sign of the square root is chosen so that p > 0.

3.2-2. Parametric Representation of a Plane. The parametric representation (Sec.3.1-14) of any plane has the form

3.3. THE STRAIGHT LINE

3.3-1. Equations of the Straight Line. (a) Two linearly independent linear equations

(A₁ X A₂ ≠ 0; see also Secs.1.9-3a, 5.2-2, and 5.2-7) represent a straight line (intersection of two planes, Sec.3.1-16a). Conversely, every straight line situated in the finite portion of space can be represented in the form (1). Equation (1) represents a straight line through the origin if and only if D₁ = D₂ = 0.

(b) The following special forms of the equations of a straight line are of interest:

c) The quantities

are direction numbers (Sec.3.1-8b) of the straight line described by Eq. (1) [direction numbers of the line of intersection of the two planes (1)].The

corresponding direction cosines cos α_x, cos α_y,cos α_Zand thus the angles α_x, α_y, α_z between the straight line and the x, y, and z axes, are given by

The (arbitrary) sign of the square root determines the positive direction of the straight line.

Equations (2) and (3) may also be interpreted as yielding the direction numbers and direction cosines of the straight line normal to two directions described by their respective direction numbers A₁, B₁, C₁,and A₂,B₂,C₂.

(d) The equations of the planes projecting the straight line (1) onto the xy plane, the xz plane, and the yz plane (i.e., planes through the straight line and perpendicular to the respective coordinate planes) are, respectively,

Any two of the equations (4) describe the straight line (1).

3.3-2. Parametric Representation of a Straight Line. The rectangular cartesian coordinates x, y, z of a point on a straight line satisfy the parametric equations (Sec.3.1-13)

[straight line through the point (ri) in the direction of the vector a].

3.4. RELATIONS INVOLVING POINTS, PLANES, AND STRAIGHT LINES

3.4-1. Angles. (a) The angle γ₁ between two straight lines having the direction cosines cos α_x, cos α_y, cosα_zandcos α′_x, cos α′_y, cosα′_z is given by (see also Sec.3.1-8c)

If the straight lines are given in the parametric form (Sec. 3.3-2) r = r′₁ + t_a′,

The two straight lines are parallel if cos γ₁ = 1 and mutually perpendicular if cos γ₁ = 0.

(b) The angle γ₂ between (the normals of) two planes A_x + B_y + C_z + D = 0 and A'x + B'y + Cz + D' = 0 or A . r + D = 0 and A′ . r + D′ = 0 is given by

If the planes are given in the parametric form (Sec.3.2-2) r = r₁ + u¹a + u²b,r = r′₁+ u¹a′ + u²b′ then

In particular, the two planes are parallel if cos γ₂ = 1, and mutually perpendicular if cos γ₂ = 0.

and (its projection on) the plane Ax + By + Cz + D — 0 is given by

In particular, the straight line is parallel to the plane if sin γ₃ = 0 (and lies in the plane if, in addition, Ax₁ + Bx₁ + Cx₁ + D — 0) and perpendicular to the plane if sin γ₃ = 1. The angle between the straight line and the normal to the plane equals 90 deg - γ₃.

3.4-2.Distances (a) Distance d₀ between the point (x₀, y₀, z_o)= (r₀) and the plane Ax + By + Cz + D = 0 or A . r + D = 0:

where the sign of the square root is chosen to be opposite to that of D. d_o is positive if the plane lies between the origin and the point (x_o, y₀, z_o).

If the equation of the plane is given in the parametric form (Sec.3.2-2) r = r₁ + ua + vb₁,

(b) Distance d′₀ between the point (x₀, y₀, z₀) = (r_o) and the straight line

(x — x₁)/cosα_x=(y — y₁)/cos α_y = (z — z₁)/cos α_z:

where sin γ₁ is given by Eq. (2). If the lines are parallel (sin γ₁ = 0) d₁ is given by Eq. (9).

NOTE: If x₁ y₁, z₁ are replaced by variable coordinates x, y, z, Eq. (10) describes the plane through the first straight line and parallel to the second. If the two straight lines intersect, this plane contains both of them.

(d) Distance d₁ between two parallel planes Ax + By + Cz + D = 0, Ax + By + Cz + D′ = 0:

(e) The distance d₃ between a plane and a parallel straight line

is given by Eq. (7).

3.4-3. Special Conditions (a) Note the following special conditions about points:

1. Three points (x₁ y₁z₁), (x₂ y₂z₃), (x₃ y₃z₃) lie on a straight line(are collinear) if and only if

(λ≠ 0), or [see also Eq. (3.1-15);

2. Four points (x₁, y₁, z₁), (x₂, y₂,z₂), (x₃, y₃, z₃), (x₄, (x₄, y₄, z₄)lie in a plane if and only if [see also (Eq.3.1-17) and Sec.3.2-1b]

(b) Note the following special conditions about planes:

1. Three planes Ax + By + Cz + D = 0, A′x + B′y+C′z + D′ = 0, A″x+B″y+ C″z+D″=0 intersect in a straight line if

unless two of them are parallel. Equation (16) implies that the equations of the three planes are linearly dependent (Sec.1.9-3a).

2. Four planes Ax + By + Cz + D = 0, A′x + B′y+C′z + D′ = 0, A″x+B″y+ C″z+D″=0, A′″x + B′″y+C′″z + D′″=o intersect in a point (or are parallel) if and only if

i.e., if the equations of the four planes are linearly dependent (Sec.1.9-3a).

1. Two straight lines lie in a plane (i.e., they intersect or are parallel) if and only if four planes determining them intersect in a point [Eq. (17)].

2. Three straight lines intersect in a point (or at least two of them are parallel) if every pair of the three lies in a plane.

3.4-4. Plane Coordinates and the Principle of Duality. The equation

describes a plane (ξ, η, ζ) “labeled” by the plane coordinates ξ, η, ζ. If the point coordinates x, y, z are considered as constant parameters and the plane coordinates ξ, η, ζ as variables, Eq. (18) may be interpreted as the equation of the point (x, y_} z) [point of intersection of all planes (18)]. Note the principle of duality (see also Sec.2.3-3): to every theorem involving only the relative positions of points, planes, and straight lines, there corresponds another theorem obtained by interchanging the terms “point” and “plane” in the original theorem.

3.4-5. Miscellaneous Relations.

(a) If φ(x, y, z)= 0 and φ(x, y, z) = 0 are the equations of two planes, the equation

describes a plane through their line of intersection (or parallel to both if they are parallel). If the equations of the first two planes are given in the normal form (Sec.3.2-1b), then —λ is the ratio of the respective distances (7) between the first and second plane and any one point on the third plane; the planes (19) corresponding to X = 1 and X = —1 bisect the angles between the given planes.

(b) The equation of the normal through the point (#o, 2/o, So) to the plane Ax + By + Cz + D = 0 is

(c) Direction numbers and direction cosines of the line of intersection of two planes are given by Eqs. (3.3-2) and (3.3-3).

(d) The point of intersection of three planes Ax + By + Cz + D = 0,

A'x + B'y + C'z + D' = 0, A“x + B“y + C“z + D“ = 0 has the coordinates

which may also be interpreted as the coordinates of the point of intersection (piercing point) of a plane and a straight line.

3.5. QUADRIC SURFACES

3.5-1. General Second-degree Equation. The following sections deal with the quadric surfaces represented by the general second-degree equation

In vector form, Eq. (1) becomes

where the tensor A has the components A_kⁱ = a_ik, and the vector a has the components a_i = a_i₄ (see also Sec.16.9-2).

3.5-2. Invariants. For any equation (1), the four quantities

and the signs of the quantities *

FIG. 3.5.1. Proper quadric surfaces.

Table 3.5-1. Classification of Quadric Surfaces (Quadrics)

are invariants with respect to the translation and rotation transformations (3.1-19), (3.1-20), and (3.1-23) and define properties of the quadric which do not depend on position. The determinant A is called the discriminant of Eq. (1).

3.5-3. Classification of Quadrics. Table 3.5-1 shows the classification of quadric surfaces in terms of the invariants defined in Sec.3.5-2.

3.5-4. Characteristic Quadratic Form and Characteristic Equation. Important properties of quadric surfaces may be studied in terms of the (symmetric) characteristic quadratic form

corresponding to Eq. (1). In particular, a proper central quadric (A ≠ 0, D ≠ 0) is a real ellipsoid, imaginary ellipsoid, or hyperboloid if F₀(x, y, z) is, respectively, positive definite, negative definite, or indefinite as determined by the (necessarily real) roots λ₁, λ₂, λ₃ of the characteristic equation (Sec.13.4-5a)

λ₁, λ₂, λ₃are the eigenvalues of the matrix corresponding to the quadratic form (5) (Sec.13.4-2).

3.5-5. Diametral Planes, Diameters, and Centers of Quadric Surfaces, (a) A diametral plane of a (proper or improper) quadric surface described by Eq. (1) is the locus of the centers of parallel chords. The diametral plane conjugate to (the direction of) the chords having the direction cosines cos _x, cos a_y, cos a_z bisects these chords and has the equation

(b) The intersection of two diametral planes is called the diameter conjugate to the family of planes paralleling the conjugate chords of both diametral planes. The diameter conjugate to the planes whose normals have the direction cosines cos a_x, cos a_y, cos a_z (Sec.3.2-1b) is represented by

(c) All diameters of a quadric (1) either intersect at a unique point, the center of the quadric (see Sec. 3.5-8d for an alternative definition), or they are parallel, according to whether D ≠ 0 or D = 0. In the former case, the quadric is a central quadric.

The coordinates x₀ y₀, z₀ of the center are given by

so that

Given the equation (1) of a central quadric, a translation (3.1-19) of the coordinate origin to the center (10) of the quadric results in the new equation

(d) Three diameters of a central quadric are called conjugate diameters if and only if each of them is the diameter conjugate to the plane of the two others.

3.5-6. Principal Planes and Principal Axes (a) A diametral plane perpendicular to its conjugate chords is a plane of symmetry or principal plane (principal diametral plane) of the quadric; the chords conjugate to a principal plane are called principal chords. Every quadric surface has at least two mutually perpendicular principal planes. A central quadric (Sec.3.5-5c) has at least three mutually perpendicular principal planes.

(b) The diameter along the line of intersection of two principal planes is a principal axis (symmetry axis). Every quadric surface has at least one principal axis; if it has more than one, there exists at least one other principal axis perpendicular to each. A central quadric has at least three mutually perpendicular principal axes, which are necessarily conjugate diameters normal to three corresponding principal planes.

The directions of the normals to the principal planes of a quadric (1), and hence also such principal axes as may exist, are directed along the eigenvectors associated

Table 3.5-2. Equations (in Standard or Type Form) and Principal Properties of Proper Quadric Surfaces

(see also Table 3.5-1; refer to Sec.3.5-8 for equations of tangent and polar planes)

with the matrix [a_ik] (Sec.14.8-6). Their direction cosines cos α_x ;cos α_y,cos α_zsatisfy the conditions

where λ is a root of the characteristic equation (6).

3.5-7. Transformation of the Equation of a Quadric to Standard Form.If a new reference system is introduced by combining

1. A rotation of the coordinate axes (Sec.3.1-12b) such that each new coordinate axis is directed either along, or perpendicular to, one of the principal-axis directions specified in Sec.3.5-6 (principal-axis transformation, Sec. 14.8-6)

2. A suitable translation (Sec.3.1-12a) of the origin

it is possible to reduce the equation (1) of any proper quadric to one of the standard or type forms (sometimes called canonical forms) listed in Table 3.5-2. Table 3.5-2 also shows the principal properties of the individual surfaces as well as the relations between the parameters a², b² c² appearing in the standard forms to the invariants A, D, J, and I of the general equation (1) (Sec.3.5-2).

The equations of the individual improper quadrics (Table 3.5-1) may be similarly reduced to the standard or type forms presented below:

In general, the directions of the new x, y, z axes are determined by Eq. (12) except for rotations through integral multiples of 90 deg about any of the new axes, corresponding to interchanges between any of the quantities x —x,y —y, z—z in the standard forms of Table 3.5-2 and Eq. (13).

3.5-8. Tangent Planes and Normals of Quadric Surfaces; Polar Planes and Poles. (a) Refer to Sec.17.3-2 for the definitions of tangent planes and normals of suitable three-dimensional surfaces. The equation of the tangent plane to the general quadric (1) at the point x₁, y₁, z₁) of the quadric is

(b) The equations of the normal to the quadric (1) at the point (x₁, y₁, z₁) are (see also Sec. 3.3-1b)

(c)No matter whether the point (x₁ y₁ z₁) in Sec.3.5-8b does or does not lie on the surface (1), Eq. (14) defines a plane called the polar plane of the pole (x₁ y₁ z₁) with respect to the quadric (1). The polar plane of a point on the quadric surface is the tangent plane at that point.

(d)The center of a quadric is the pole of the plane at infinity (a common definition of the center; if this definition is used, diametral planes and diameters are defined as planes and straight lines, respectively, through the center).

(e)The polar planes of all points of any given plane pass through the pole of the given plane.

(f)The polar planes of all points of a given straight line pass through a straight line. Conversely, the latter contains the poles of all planes passing through the first straight line.

(g)The tangents from any point to a quadric generate an elliptic cone or a pair of planes; the points of tangency lie on the polar plane of the given point.

3.5-9. Miscellaneous Formulas and Theorems Relating to Quadric Surfaces.

(a) Equation of a sphere of radius r about the point (x_h y_h Zi)

(b) The most general form of the equation of a sphere is

(c) Let P₁ and P₂ be the points of intersection of a sphere and a straight line (secant) through a fixed point P₀ = (x_o, y_o, z₀). Then the products of the directed line segments and are equal for all secants through P₀, and

The theorem holds also if P₁ and P₂ coincide (i.e., if the secant becomes a tangent).

(d) Given any three conjugate semidiameters (Sec.3.5-5d) of a given ellipsoid,

1. The sum of their squares is a² + b² + c².

2. The parallelepiped having them as adjacent sides has the volume abc.

is the equation of an ellipsoid, hyperboloid of one sheet, or hyperboloid of two sheets (see also Table 3.5-2), then

is the equation of the diameter conjugate to the plane whose normal has the direction cosines cosα _x,cos α_y cosα _z. The respective direction cosines cosα _x,cosα _y,cosα _z; cosβ_x, cos β_y, cosβ _z,cosγ _x,cosγ _y,cosγ _zof three conjugate diameters satisfy the rolafinna

The signs in Eqs. (20) and (21) correspond to those in Eq. (19).

3.5-10. Parametric Representation of Quadrics (see also Secs.3.1-14 and 6.5-1). Note the following parametric representations of quadrics:

Many other parametric representations are possible.

Algebraic equations, determinants Chap. 1

Plane analytic geometry Chap. 2

Coordinate transformations Chaps. 2, 6 14, 16

Curvilinear coordinate systems Chap. 6

Matrices Chap. 13

Differential geometry Chap. 17

Mensuration formulas Appendix A

Plane and spherical trigonometry Appendix B

3.6-2. References and Bibliography. Refer to Sec.2.7-2.

* A_ik denotes the cofactor of a_{i_k} in the fourth-order determinant A — det |a,ik| (Sec.1.5-2).