Mathematical Handbook for Scientists and Engineers

2.1-1. Introduction (see also Sec. 12.1-1). A geometry is a mathematical model involving relations between objects referred to as points. Each geometry is defined by a self-consistent set of defining postulates; the latter may or may not be chosen so as to make the properties of the model correspond to physical space relationships. The study of such models is also called geometry. Analytic geometry represents each point by an ordered set of numbers (coordinates), so that relations between points are represented by relations between coordinates.

Chapters 2 (Plane Analytic Geometry) and 3 (Solid Analytic Geometry) introduce their subject matter in the manner of most elementary courses: the concepts of Euclidean geometry are assumed to be known and are simply translated into analytical language. A more flexible approach, involving actual construction of various geometries from postulates, is briefly discussed in Chap. 17. The differential geometry of plane curves,including the definition of tangents, normals, and curvature, is outlined in Secs. 17.1-1 to 17.1-6.

FIG. 2.1-1. Right-handed oblique cartesian coordinate system. The points marked “1” define the coordinate scales used.

2.1-2. Cartesian Coordinate Systems. A cartesian coordinate system (cartesian reference system, see also Sec. 17.4-6b) associates a unique ordered pair of real numbers (cartesian coordinates), the abscissa x and the ordinate y, with every point P ≡ (x, y)in the finite portion of the Euclidean plane by reference to a pair of directed straight lines (coordinate axes) OX, OY intersecting at the origin 0 (Fig.2.1-1). The parallel to OY through P intersects the x axis OX at the point P′. Similarly, the parallel to OX through P intersects the y axis OY at P″.

The directed distances OP′ = x (positive in the positive x axis direction) and OP″ = y (positive in the positive y axis direction) are the cartesian coordinates of the point P ≡ (x, y).

x and y may or may not be measured with equal scales. In a general (oblique) cartesian coordinate system, the angle XOY =ω between the coordinate axes may be between 0 and 180 deg (right-handed cartesian coordinate systems) or between 0 and –180 deg (left-handed cartesian coordinate systems).

A system of cartesian reference axes divides the plane into four quadrants (Fig.2.1-1). The abscissa x is positive for points (x, y) in quadrants I and IV, negative for points in quadrants II and III, and zero for points on the y axis. The ordinate y is positive in quadrants I and II, negative in quadrants III and IV, and zero on the x axis. The origin is the point (0, 0).

NOTE: Euclidean analytic geometry postulates a reciprocal one-to-one correspondence between the points of a straight line and the real numbers (coordinate axiom, axiom of continuity, see also Sec. 4.3-1).

2.1-3. Right-handed Rectangular Cartesian Coordinate Systems.

FIG. 2.1-2. Right-handed rectangular cartesian coordinate system and polar-coordinate system.

In a right-handed rectangular cartesian coordinate system, the directions of the coordinate axes are chosen so that a rotation of 90 deg in the positive (counterclockwise) sense would make the positive x axis OX coincide with the positive y axis 0 Y (Fig. 2.1-2). The coordinates x and y are thus equal to the respective directed distances between the y axis and the point P, and between the x axis and the point P.

Throughout the remainder of this chapter, all cartesian coordinates x,y refer to right-handed rectangular cartesian coordinate systems, and equal scale units of unit length are used to measure x and y, unless the contrary is specifically stated.

2.1-4. Basic Relations in Terms of Rectangular Cartesian Coordinates. In terms of rectangular cartesian coordinates (x,y)the following relations hold:

1. The distance d between the points P₁ ≡ (x₁, y₁)and P2 ≡ (x₂, y₂) is

The oblique angle γ between two directed straight-line segments and is given by

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

The direction cosines cos α_x and cos α_y of a directed line segment are the cosines of the angles α_x and α_y = 90 deg – α_xrespectively.

3.The coordinates x, yof the point P dividing the directed line segment between the points P₁ ≡ (x₁, y₁) and P₂ ≡ (x₂, y₂) in the ratio =m:n = μ :1 are

4. The area S of the triangle with the vertices P₁ ≡ (x₁, y₁), P₂ ≡ (x₂,y₂),P₃≡ (x₃, y₃) is

This expression is positive if the circumference P₁P₂P₃ runs around the inside of the triangle in a positive (counterclockwise) direction. Specifically, if x₃ = y₃ = 0,

2.1-5. Translation of the Coordinate Axes. Let x, y 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 those of the x, y system, and whose origin has the coordinates x = x₀and y = y₀ in the x, ysystem. If equal scales are used to measure the coordinates in both systems, the coordinates are related to the coordinates x,yby the transformation equations (Fig.2.1-3a; see also Chap. 14)

The equations (8) permit a second interpretation. If are considered as coordinates referred to the x, y system of axes, then the point defined by is translated by a directed amount – x₀ in the x axis direction and by a directed amount – y ₀ in the y axis direction with respect to the point (x, y). Transformations of this type applied to each point x,y of a plane curve may be used to indicate the translation of the entire curve.

2.1-6. Rotation of the Coordinate Axes. Let x , y 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 having the same origin 0 and rotated with respect to the x, ysystem so that the angle XOX̄ between the x axis OX̄ and axis OX̄ is equal to ϑ measured in radians in the positive (counterclockwise) sense (Fig.2.1-3b). If equal scales are used to measure all four coordinates the coordinates x, y, , the coordinates are related to the coordinates z, y by the transformation equations

A second interpretation of the transformation (9) is the definition of a point () rotated about the origin by an angle –ϑ with respect to the point (x, y).

2.1-7. Simultaneous Translation and Rotation of Coordinate Axes. If the origin of the system in Sec. 2.1-6 is not the same as the origin of the x, ysystem but has the coordinates x = x₀ and y = y_o in the x, ysystem, the transformation equations become

FIG.2.1-3a. Translation of coordinate axes.

FIG.2.1-3b. Rotation of coordinate axes.

The relations (10) permit one to 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.

The transformation (10) may also be considered as the definition of a point translated and rotated with respect to the point (x, y).

NOTE: The transformations (8), (9), and (10) do not affect the value of the distance (1) between two points or the value of the angle given by Eq. (2). The relations constituting Euclidean geometry are unaffected by (invariant with respect to) translations and rotations of the coordinate system (see also Secs. 12.1-5, 14.1-4, and 14.4-5).

2.1-8. Polar Coordinates. A plane polar-coordinate system associates ordered pairs of numbers r, φ (polar coordinates) with each point P of the plane by reference to a directed straight line OX (Fig.2.1-2), the polar axis. Each point P has the polar coordinates r, defined as the directed distance OP, and φ, defined as the angle XOP measured in radians in the counterclockwise sense between OX and OP. The point 0 is called the pole of the polar-coordinate system; r is the radius vector of the point P.

Negative values of the angle φ are measured in the clockwise sense from the polar axis OX. Points (r, φ) are by definition identical to the points (–r, φ ± 180 deg); this convention associates points of the plane with pairs of numbers (r, φ) with negative as well as positive radius vectors r.

NOTE: Unlike a cartesian coordinate system, a polar-coordinate system does not establish a reciprocal one-to-one correspondence between the pairs of numbers (r, φ) and the points of the plane. The ambiguities involved may, however, be properly-taken into account in most applications.

If the pole and the polar axis of a polar-coordinate system coincide with the origin and the x axis, respectively, of a right-handed rectangular cartesian coordinate system (Fig. 2.1-2), then the following transformation equations relate the polar coordinates (r, φ) and the rectangular cartesian coordinates (x, y) of corresponding points if equal scales are used for the measurement of r, x, and y:

In terms of polar coordinates (r, φ), the following relations hold:

1. The distance d between the points (r₁,φ₁)and (r₂, φ ₂) is

2. The area S of the triangle with the vertices P₁ ≡ (r₁, φ ₁), P₂ ≡ (r₂, φ ₂), and P ₃≡ (r₃, φ ₃), is

This expression is positive if the circumference P₁P₂P₃runs around the inside of the triangle in the positive (counterclockwise) direction. Specifically, if r₃= 0,

See Chap. 6 for other curvilinear coordinate systems.

2.1-9. Representation of Curves (see also Secs. 3.1-13 and 17.1-1).

(a) Equation of a Curve. A relation of the form

is, in general, satisfied only by the coordinates x, y of points belonging to a special set defined by the given relation. In most cases of interest, the point set will be a curve (see also Sec. 3.1-13). Conversely, a given curve will be represented by a suitable equation (15), which must be satisfied by the coordinates of all points (x,y)on the curve. Note that a curve may have more than one branch.

The curves corresponding to the equations

where λ is a constant different from zero, are identical.

(b) Parametric Representation of Curves. A plane curve can also be represented by two equations

where t is a variable parameter.

(c) Intersection of Two Curves. Pairs of coordinates x,ywhich simultaneously satisfy the equations of two curves

represent the points of intersection of the two curves. In particular, if the equation φ(x, 0) = 0 has one or more real roots x, the latter are the abscissas of the intersections of the curve φ(x, y) = 0 with the x axis.

If φ (x, y) is a polynomial of degree n (Sec. 1.4-3), the curve φ (x, y) = 0 intersects the x axis (and any straight line, Sec. 2.2-1) in n points (nth-order curve); but some of these points of intersection may coincide and/or be imaginary.

For any real number λ, the equation

describes a curve passing through all points of intersection (real and imaginary) of the two curves (17).

(d) Given two curves corresponding to the two equations (17), the equation

is satisfied by all points of both original curves, and by no other points

2.2. THE STRAIGHT LINE

2.2-1. The Equation of the Straight Line. Given a right-handed rectangular cartesian coordinate system, every equation linear in x and y, i.e., an equation of the form

where A and B must not vanish simultaneously, represents a straight line (Fig.2.2-1). Conversely, every straight line in the finite portion of the plane can be represented by a linear equation (1). The special case C = 0 corresponds to a straight line through the origin.

The following special forms of the equation of a straight line are of particular interest:

When the equation of a straight line is given in the general form (1), the intercepts a and b, the slope m, the perpendicular distance p from the origin, cos ϑ, and sin ϑ are related to the parameters A, B, and C as follows:

In order to avoid ambiguity, one chooses the sign of in Eq. (3) so that p > 0.

In this case, the directed line segment between origin and straight line defines the direction of the positive normal of the straight line; cos ϑ and sin ϑ are the direction cosines of the positive normal (Sec. 17.1-2).

2.2-2. Other Representations of Straight Lines. In terms of a variable parameter t ,the rectangular cartesian coordinates x and y of a point on any straight line may be expressed in the form

If the sign of is chosen so that p > 0, the origin will always lie on the right of the direction of motion as t increases, i.e., in the direction of the negative normal (Sec. 17.1-2).

In terms of polar coordinates r, φ, the equation of any straight line may be expressed in the form

where A, B, C, p and ϑ are defined as in Sec. 2.2-1.

2.3. RELATIONS INVOLVING POINTS AND STRAIGHT LINES

2.3-1. Points and Straight Lines. The directed distance d from the straight line (2.2-1) to a point (x₀, y₀ is

where the sign ofis chosen to be opposite to that of C. d is positive if the straight line lies between the origin and the point (x₀, y₀).

Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are on a straight line if and only if (see also Sec. 2.2-1)

2.3-2. Two or More Straight Lines. (a) Two straight lines

intersect in the point

(b) Either angle γ ₁₂ from the straight line (3a) to the line (3b) is given by

where γ ₁₂ is measured counterclockwise from the line (3a) to the line (3b).

(c) The straight lines (3a) and (3b) are parallel if

and perpendicular to each other if

(d) The equation of a straight line through a point (x₁, y₂)and at an angle γ ₁₂ (or 180 deg – γ ₁₂) to the straight line (3a) is

Specifically, the equation of the normal to the straight line (3a) through the point (x₁, y₁) is

(e) The equation of any straight line parallel to (3a) may be expressed in the form

The distance d between the parallel straight lines (3a) and (10) is

If the sign of in Eq. (11) is chosen to be the opposite to that of C₁ will be positive if the straight line (3a) is between the origin and the straight line (10).

(f) The equation of every straight line passing through point of inter-section of two straight lines (3a) and (3b) will be of the form

with λ ₁, λ ₂ not both equal to zero. Conversely, every equation of the form (12) describes a straight line passing through the point of intersection. If the straight lines (3a) and (3b) are parallel, Eq. (12) represents a straight line parallel to the two. If the straight lines are given in the normal form (Sec. 2.2-1), – λ is the ratio of the distances (1) between the first and second straight line and any one point on the third straight line, and the straight lines corresponding to λ = 1 and λ = – 1 bisect the angles between the given straight lines.

(g) Three straight lines

intersect in a point or are parallel if and only if

i.e., if the three equations (13) are linearly dependent (Sec. 1.9-3a).

2.3-3. Line Coordinates. The equation

describes a straight line (ξ, η) “labeled” by the line coordinates ξ, η. If the point coordinates x, y are considered as constant parameters and the line coordinates ξ, η as variables, Eq. (15) may be interpreted as the equation of the point (x, y) [point of intersection of all straight lines (15)]. The symmetry of Eq. (15) in the pairs (x, ξ) and (yt ηj) results in a correspondence (duality) between theorems dealing with the positions of points and straight lines (Secs. 2.3-1 and 2.3-2). An equation F(ξ, η) = 0 represents a set of straight lines which will, in general, envelop a curve determined by the nature of the function F(ξ, η) (see also Table2.4-2).

2.4. SECOND-ORDER CURVES (CONIC SECTIONS)

2.4-1. General Second-degree Equation. The second-order curves or conic sections (conics) are represented by the general second-degree equation

2.4-2. Invariants. For any equation (1), the three quantities

and the sign of quantity

are invariants with respect to the translation and rotation transformations (2.1-8), (2.1-9), and (2.1-10). Such invariants define properties of the conic which do not depend on its position.

Either A or Δ = 8A is sometimes called the discriminant of Eq. (1).

2.4-3. Classification of Conics. Table 2.4-1 shows the classification of conics in terms of the invariants defined in Sec. 2.4-2.

2.4-5. Characteristic Quadratic Form and Characteristic Equation. Important properties of conics may be studied in terms of the (symmetric) characteristic quadratic form

corresponding to Eq. (1). In particular, a proper conic (A ≠ 0) is a real ellipse, imaginary ellipse, hyperbola, or parabola if F₀ (x, y) is, respectively, positive definite, negative definite, indefinite, or semidefinite as determined by the (necessarily real) roots λ ₁, λ₂ of the characteristic equation

λ₁ and λ₂ are the eigenvalues of the matrix a_ik (Secs. 13.4-5 and 13.5-2).

2.4-6. Diameters and Centers of Conic Sections, (a) A diameter of a conic described by Eq. (1) is the locus of the centers of parallel chords. The diameter conjugate to the chords inclined at an angle ϑ with respect to the positive x axis bisects these chords and is a straight line with the equation

(b) All diameters of a conic (1) either intersect at a unique point, the center of the conic (see Sec. 2.4-10 for an alternative definition), or they are parallel, according to whether D ≠ 0 or D = 0. In the former case, the conic is a central conic.

The coordinates x₀ y₀ of the center are given by

Given the equation (1) of a central conic, a translation (2.1-8) of the coordinate origin to the center (8) of the conic results in the new equation

in terms of the new coordinates

(c) Two conjugate diameters of a central conic each bisect the chords parallel to the other diameter (see also Sec. 2.5-2e).

2.4-7. Principal Axes. A diameter perpendicular to its associated (conjugate) chords (principal chords) is a symmetry axis or principal axis of the conic. Every (real) noncentral conic has one principal axis; every (real) central conic has either two mutually perpendicular principal axes, or every diameter is a principal axis (circle).

The principal axes are directed along eigenvectors of the matrix [a_ik] (Sec. 14.8-6). More specifically, the direction cosines cos ϑ, sin ϑ of the normal to a principal axis (Sec. 2.2-1) satisfy the conditions

where λ is a nonvanishing root of the characteristic equation (5). The angle Φ between the positive x axis and any principal axis of the conic (1) satisfies the condition

2.4-8. Transformation of the Equation of a Conic to Standard or Type Form. If one introduces a new reference system by combining a rotation of the coordinate axes through an angle satisfying Eq. (11) with a suitable translation of the origin (Sec. 2.1-7), the equation (1) of any proper conic reduces to one of the standard or type forms (sometimes called canonical forms) listed below. The parameters a², b², pappearing in the standard forms are simply related to the invariants A, D, I, and to the roots λ ₁ ≥ λ ₂ of the characteristic equation (5).

(2.4-12a)(2.4-12b)(2.4-12c)

The equations of the improper (degenerate) conics are similarly transformed to the standard or type forms

NOTE: Every rotation (2.1-9) through an angle ϑ satisfying Eq. (11) diagonalizes the matrix [a_ik] of the characteristic quadratic form (4) (principal-axis transformation, see also Sec. 14.8-6). The values of ϑ satisfying Eq. (11) differ by multiples of 90 deg, corresponding to interchanges of x, – x, y, and –y. The standard forms (12a, b, c) correspond to choices of ϑ such that the foci (Sec. 2.4-9) of the conics lie on the x axis. Equation (11) becomes indeterminate for (real and imaginary) circles and points, which have no definite principal axes.

2.4-9. Definitions of Proper Conics in Terms of Loci. Once the equation of any proper conic is reduced to its standard form (12), a simple translation (Sec. 2.1-5) may be used to introduce a new system of coordinates x,ysuch that the equation of the conic appears in the form

The conic passes through the origin of the new x, y system; the x axis is a symmetry axis(principal axis) of the conic.

Equation (14) describes a proper conic as the locus of a point which moves so that the ratio ≥ 0 (eccentricity) of its distances from a fixed point (focus) and from a fixed line (directrix) is a constant. The conic will be an ellipse if 1 and, specifically, a circle if ∊ = 0. The conic will be a hyperbola if 1, and a parabola if ∊ = 1.

The equation of the directrix of the conic represented by Eq. (14) is

The coordinates x and y of the focus are

The directrix is perpendicular to the symmetry axis. The latter passes through the focus and also through the vertex x = y = 0 of the conic. The distance between the focus and the directrix is equal to 2p/.

In the case of a central conic (ellipse or hyperbola), the straight line

is a symmetry axis (principal axis) of the conic, so that two foci and two directrices can be defined.

The latus rectum of a proper conic is defined as the length of a chord through the focus and perpendicular to the symmetry axis and is equal to |4p|.

NOTE : All types of improper as well as of proper conics may be obtained as the inter-sections of a right circular cone with a plane for various inclinations of the plane with respect to the cone. If the conic is a pair of (distinct, coincident, or imaginary) parallel straight lines (see also Table 2.4-1), then the cone must be regarded as degenerated into a cylinder unless the plane is tangent to the cone.

2.4.10. Tangents and Normals of Conic Sections. Polars and Poles. The equation of the tangent (Sec. 17.1-1) to the general conic (1) at the point (x₁,y₁) of the conic is

The equation of the normal (Sec. 17.1-2) to the conic (1) at the point (x₁,y₁)is

Equation (18) defines a straight line called the polar of the pole (x₁, y₁) with respect to the conic (1), whether the point (x₁, y₁)does or does not lie on the curve. The polar of a point on the conic is the tangent at that point.

The equations of tangents, polars, and normals for proper conic sections described in the standard form (Sec. 2.4-8) are listed in Table 2.4-2.

NOTE the following theorems about polars and poles with respect to conic sections.

1. If the pole moves along a straight line, the corresponding polar rotates about the pole of the straight line, and conversely.

2. If two tangents can be drawn from a point to a conic section, the polar of the point passes through the points of contact.

3. For any straight line drawn through the pole P and intersecting the polar at Q and the conic section at R₁ and R₂, the points P and Q divide R₁R₂ harmonically, i.e.

The tangents at the end points of any chord through a focus intersect on the corresponding directrix.

Every chord through a focus is perpendicular to the line drawn through the focus and the point of intersection of the tangents at the end points of the chord.

The tangent at an end of a diameter is parallel to the chords defining the diameter.

The tangents drawn at the ends of any chord intersect on the diameter conjugate to the chord.

The pole of any diameter is the point at infinity. The polar of any point on a diameter is parallel to the chords defining the diameter. The center of a conic is the pole of the straight line at infinity (a common definition of the center; if this definition is used, diameters are defined as straight lines through the center).

2.4.11. Other Representations of Conics, (a) A conic is definitely determined by five of its points if no four of them are collinear (Sec. 2.3-1). The equation of the conic through five such points (x₁, y₁), (x₂, ₂), (x₃, y₃), (x₄, y₄), (x₅, y₅) is

The conic will be improper if and only if any three of the given points are collinear. Equation (20) may also be interpreted as a condition that six points lie on a conic.

NOTE: The construction of a conic through five given points is made possible by Pascal's theorem: For any closed hexagon whose vertices lie on a conic, the intersections of opposite pairs of sides are collinear (or at infinity).

A conic is also definitely determined by five tangents, if no four of them intersect in a point; Brianchon's theorem states that for any hexagon whose sides are tangent to a conic, the diagonals connecting opposite vertices intersect in a point (or are parallel).

(b) If the focus of a proper conic is chosen as the pole and the symmetry axis as the polar axis of a polar-coordinate system, the equation of the conic in terms of the polar coordinates r, φ is

2.5. PROPERTIES OF CIRCLES, ELLIPSES, HYPERBOLAS, AND PARABOLAS

2.5.1. Special Formulas and Theorems Relating to Circles (see also Table 2.4-2). (a) The general form of the equation of a circle in rectangular cartesian coordinates is

The point (x₀, y₀) is the center of the circle, and R is its radius. The circle (1) touches the x axis or the y axis if 4C = A² or 4C = B², respectively. The equation of a circle about the origin is

(b) The equation of the circle through three noncollinear points (x₁, y₁), (x₂, y₂), (x₃, y₃) is

Equation (3) is a necessary and sufficient condition that the four points (x, y), (x₁, y₁), (x₂, y₂), (x₃, y₃) lie on a circle.

(c) The length L of each tangent from a point (x₁,y₁)to the circle (1) is

(d) Two circles

are concentric if and only if A₁ = A₂ and B₁ = B₂. They are orthogonal if and only if A₁A₂ + B₁B₂ = 2(C₁ + C₂).

(e) All circles passing through the (real or imaginary) points of intersection of the two circles (5) have equations of the form (see also Sec. 2.1-9c)

where λ is a parameter. Note that the curves (6) exist even if the two circles (5) have no real points of intersection.

(f) For λ = – 1, Eq. (6) reduces to the equation of a straight line

which is called the radical axis of the two circles (5). The radical axis is the locus of points from which tangents of equal length can be drawn to the two circles. If the two circles intersect or touch, the radical axis is the secant or tangent through the common points or point. The radical axis of two concentric circles may be considered as infinitely far away from their common center. The three radical axes associated with three circles intersect in a point (radical center). This fact is used for the construction of the radical axis of two nonintersecting circles. If the centers of three circles are collinear, their radical center is at infinity.

(g) In terms of polar coordinates r, φ, the equation of a circle of radius R about the point (r₀, φ₀) is

2.5.2. Special Formulas and Theorems Relating to Ellipses and Hyperbolas (see also Sec. 2.5-3 and Tables 2.4-2 and 2.5-1). (a) For any ellipse (2.4-12a), the respective lengths of the major and minor axes

FIG. 2.5-1a

FIG. 2.5-1b

FIG. 2.5-1c

FIG. 2.5-1. Graphs of ellipse, hyperbola, and parabola in standard form (Sec. 2.4-8), showing foci, axes, and length of the latus rectum (Secs. 2.4-9 and 2.5-2) for each curve.

Table 2.5-1. Special Formulas Relating to Ellipses, Hyperbolas, and Parabolas Represented in Standard Form (see also Secs. 2.4-9 and 2.5-2)

( Fig.2.5-1a) are 2a and 2b. The sum of the focal radii r₁ and r₂ (Table 2.5-1, 6) is constant (and equal to 2a) for every point on an ellipse.

(b) A hyperbola (2.4-12b) approaches the asymptotes (Sec. 17.1-6)

The asymptotes are straight lines through the center of the hyperbola; they intersect at an angle equal to 2 arctan (b/a), which is equal to 90 deg for a = b (rectangular hyperbola).

(c) For every hyperbola (2.4-12b), the length of the transverse axis (distance between vertices, Fig.2.5-1b) is equal to 2 a . The conjugate axis is the principal axis perpendicular to the transverse axis. The distance between the intersections of the asymptotes with a tangent touching the hyperbola at a vertex equals 2b (Fig.2.5-1b). The difference between the focal radii r₁ and r₂ (Table 2.5-1, 6) is constant and respectively equal to 2a and –2a for points on either branch of a hyperbola.

(d) If d₁ and d₂ are the lengths of any two conjugate diameters (Sec. 2.4-6e) of an ellipse or hyperbola,

where m₁, m₂ are the respective slopes of the two diameters.

(e)A diameter y = mx of the hyperbola (2.4-12b) intersects the hyperbola if and only if m² < b²/a² and coincides with an asymptote if and only if m² = b²/a². An asymptote may be considered as a chord intersecting the hyperbola at infinity and identical with its own conjugate diameter.

(f)For every secant of a hyperbola, the two segments between the hyperbola and an asymptote are equal; for every tangent of a hyperbola the point of contact bisects the segment between the asymptotes. For every point of a hyperbola, the product of the distances to the asymptotes is constant.

(g)For every ellipse,the area of the triangle formed by the center and the end points of any two conjugate diameters is constant. For every hyperbola the area of the triangle formed by the asymptotes and any tangent is constant.

(h)Two ellipses or hyperbolas are similar if and only if the ratio of the two major axes, or transverse axes, respectively, is equal to the ratio of the two minor axes, or conjugate axes, respectively.

2.5.3. Construction of Ellipses and Hyperbolas and Their Tangents and Normals. (a) The following procedures may be used for the construction of an ellipse, given the lengths and location of the major and minor axes.

1. Let 0 be the center of the ellipse; let P′OP be the major axis and Q′OQ the minor axis. To construct a point of the ellipse, draw any straight line ORS through the center with OR = b and OS = a. Then draw a parallel to P'OP through R and a parallel to Q′OQ through S; these two lines will intersect in a point of the ellipse.

2. Obtain the foci F and F′ by using the relation QF = QF′ = OP = a. To construct points of the ellipse, draw any point T on the major axis between 0 and P. Two circles drawn about F and F′ with the radii P′t and PT will intersect in two points of the ellipse.

To construct a hyperbola, given the length and location of the transverse axis P′OP and the location of the foci F and F' (see Table 2.5-1 for the relation between OF = OF′, a, and b), draw any point T on the transverse axis such that OT > OP. Two circles drawn about F and F′ with the radii P′t and PT will intersect in two points of the hyperbola.

(b) The following properties of ellipses and hyperbolas are useful for the construction of tangents and normals to these curves.

1. For every ellipse (2.4-12a), the tangents drawn at a point (xi, yi)of the ellipse intersect the x axis at the same point as the tangents of the circle x² + y² = a² drawn at the points

2. The tangent and the normal at any point of an ellipse or hyperbola bisect the angles between straight lines connecting that point to the foci. This theorem implies the “focal property” of the ellipse.

3. The product of the distances between the foci and any tangent of an ellipse or hyperbola is constant and equal to b².The perpendiculars drawn from either focus to a tangent meet the latter on the circle drawn with the major axis (transverse axis) as a diameter; this theorem may be used to construct an ellipse or hyperbola by envelopment.

(c) One may draw a portion of an ellipse or hyperbola approximately by drawing the circles of curvature (Sec. 17.1-4) for the two vertices.For any ellipse or hyperbola, the centers of curvature for the vertices lie on the major axis or transverse axis, respectively, and the radius of curvature at either vertex is equal to b²/a.The centers of curvature for the points at either end of the minor axis of an ellipse lie on the minor axis, and the radius of curvature at either of these points is equal to a²/b.

2.5.4. Construction of Parabolas and Their Tangents and Normals.(a) The following properties of parabolas may be used for the construction of a parabola when the (directed) axis, the focus, and the distance 2p between the focus and the directrix are given:

1. The distance between the focus and any point P of a parabola is equal to the distance between the directrix and the point P (see also Sec. 2.4-9).

2. A straight line perpendicular to the axis of the parabola at a distance p from the focus in the direction of the negative axis is the tangent of the parabola at the vertex. For any point Q on this line, the perpendicular at Q to a line joining Q with the focus will be a tangent of the parabola (construction by envelopment). Conversely, any tangent and a line perpendicular to it through the focus intersect on the tangent at the vertex.

(b) The following properties of parabolas are useful in connection with the construction of tangents and normals to a parabola:

1. The distance between any point P of a parabola and the focus is equal to the distance between the focus and the intersection of the parabola axis with the tangent at P.

2. The tangent and normal at any point P of a parabola bisect the angles formed by the line joining P with the focus and the diameter through P. Note that the diameter is parallel to the axis; this theorem implies the “focal property” of the parabola.

3. The normal at any point P of a parabola and the perpendicular dropped from P on the axis intersect the latter a constant distance 2p apart.

4. For every parabola, the directrix is the locus of the intersections of pairs of tangents which are perpendicular to each other.

(c) One may draw a portion of a parabola approximately by drawing the circle of curvature (Sec. 17.1-4) for the vertex. The center of curvature for the vertex lies on the axis, and the radius of curvature at the vertex is equal to 2p.

2.6. HIGHER PLANE CURVES

2.6-1. Examples of Algebraic Curves (see also Fig.2.6-1 and Sec. 2.1-9c).

(a)Neil's parabola: y = ax^3/2

(b)Witch of Agnesi: x²y = 4a² (2a - y)

(c)Conchoid of Nicomedes:(x² + y²)(x - a)² = x²b²

(d) Cissoid of Diodes:

(e) Lemniscate of Bernoulli:

(f) Ovals of Cassini: (x² - y² + a²)² - 4a²x² = c⁴

(locus of points for which the product of the distances from (– a, 0) and (a, 0) is equal to c²)

(g) Strophoid: x^z + x(a* + y²) = 2a(y² -f x²)

(h) Cruciform:

(i) Cardioid:

(j)Trisectrix:

(k) Astroid: a^2/3 + y^2/3 = a^2/3

(1) Leaf of Descartes: x^z -\- y³ – Zaxy

(m) Limacon of Pascal: r = b – a cos φ

(n) Lituus: r²φ = a²

2.6-2.Examples of Transcendental Curves (see also Fig.2.6-2).

(a) Catenary:y = a/2 (e^x/a + e ^-x/a)

(b) (Linear) spiral of Archimedes: r = a₀

(c) Parabolic spiral: r²= 4pφ

(d)Logarithmic spiral: r = ae^bφ

FIG. 2.6-1. Examples of higher algebraic curves.

(e) The locus of a point (x,y)at the distance a₁ from the center of a circle of radius a rolling on the x axis is a cycloid and may be represented by

if the point (x,y)rolls with the circle.

(f) The locus of a point (x,y)at the distance a₁ from the center of a circle of radius a rolling on the outside of the circle x² + y² = b² is an epicycloid and may be represented by

if the point (x,y)rolls with the circle.

FIG. 2.6-2. Some transcendental curves.

(g) The locus of a point (x,y)at the distance a₁ from the center of a circle of radius a rolling on the inside of the circle x² + y² = b² is a hypocycloid and may be represented by

if the point (x,y)rolls with the circle.

(h) A tractrix may be represented by

2.7. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY

2.7-1. Related Topics. The following topics related to the study of plane analytic geometry are treated in other chapters of this handbook:

Formulas from plane geometry Appendix A

Solid analytic geometry Chap.3

Coordinate transformations Chap. 14

Differential geometry Chap. 17

Algebraic equations Chap. 1

Curvilinear coordinate systems Chap. 6

2.7-2. References and Bibliography.

2.1 Cell, J. W.: Analytic Geometry, 3d ed., Wiley, New York, 1960.

2.2 Middlemiss, R. R.: Analytic Geometry, 2d ed., McGraw-Hill, New York, 1955.

2.3 Purcell, E. J.: Analytic Geometry, Appleton-Century-Crofts, New York, 1958.

2.4 Smith, E. S., et al.: Analytic Geometry, Wiley, New York, 1954.

Additional Background Material

2.5 Coxeter, H. S. M.: Introduction to Geometry, Wiley, New York, 1961.

2.6 Klein, F.: Famous Problems of Elementary Geometry, 2d ed., Dover, New York, 1956.