ClipArt ETC

This mathematics ClipArt gallery offers 28 images of conic sections, or conics, creating hyperbolas. Conics are obtained by taking a cone, or conical surface, and intersecting it with a plane. Conic hyperbolas are retrieved by intersecting the cones with a plane, often perpendicular to the circles created by the cones. The result are unbound (meaning not closed shapes) curves that are hyperbolas.

Sections of a cone. a, parabola; b, ellipse; c, hyperbola.

Cone

Sections of a cone. a, parabola; b, ellipse; c, hyperbola.

Conjugate diameters perpendicular to each other are called, axes, and the points where they cut the curve vertices of the conic.

Conic Axes

Conjugate diameters perpendicular to each other are called, axes, and the points where they cut the…

Illustration of the rolling of equal hyperbolas. If two equal hyperbolas are placed so that the distances between their foci O<SUB>1</SUB> and O<SUB>2</SUB>, and d and e, are each equal to fg=hk, they will make contact at some point c.

Rolling of Equal Hyperbolas

Illustration of the rolling of equal hyperbolas. If two equal hyperbolas are placed so that the distances…

One of the three species of conic sections is the hyperbola.

Hyperbola

One of the three species of conic sections is the hyperbola.

Illustration showing a hyperbola as a curve formed by the intersection of the surface of a cone with a plane parallel to the axis of the cone.

Hyperbola

Illustration showing a hyperbola as a curve formed by the intersection of the surface of a cone with…

Illustration showing the asymptotes of a hyperbola. The asymptotes will never meet the curve.

Asymptotes of a Hyperbola

Illustration showing the asymptotes of a hyperbola. The asymptotes will never meet the curve.

Illustration of a hyperbola and its auxiliary circle. "Any ordinate of a hyperbola is to the tangent from its foot to the auxiliary circle as b is to a."

Auxiliary Circle and Hyperbola

Illustration of a hyperbola and its auxiliary circle. "Any ordinate of a hyperbola is to the tangent…

Diagram depicting a cone with both nappes intersected by plane J to form a hyperbola.

Cone Intersected by a Plane to Form a Hyperbola

Diagram depicting a cone with both nappes intersected by plane J to form a hyperbola.

Illustration of a cone cut by a plane parallel to the axis of the cone and perpendicular to the vertical plane of projection. The curve made by the intersection of the plane and the cone is called a hyperbola.

Conic Section Hyperbola

Illustration of a cone cut by a plane parallel to the axis of the cone and perpendicular to the vertical…

Illustration showing the definition of an hyperbola as a conic section. "The section of a right circular cone made by a plane that cuts both nappes of the cone is a hyperbola." A double right circular cone with a plane parallel to the axis cutting through the cones.

Conic Section Showing an Hyperbola

Illustration showing the definition of an hyperbola as a conic section. "The section of a right circular…

Illustration showing the definition of a hyperbola as a conic section.

Conic Section Showing Hyperbola

Illustration showing the definition of a hyperbola as a conic section.

Illustration showing the definition of an hyperbola as a conic section. A hyperbola is formed from a double right circular cone with a plane parallel to the axis cutting through the cones.

Conic Section Showing A Hyperbola

Illustration showing the definition of an hyperbola as a conic section. A hyperbola is formed from a…

An illustration showing the intersection of a plane and a double cone. The cone is intersected by a plane parallel to the axis. Thus forming a Hyperbola.

Conic Section Showing A Hyperbola

An illustration showing the intersection of a plane and a double cone. The cone is intersected by a…

Illustration showing how the a hyperbola is symmetrical with respect to its conjugate axis.

Conjugate Axis of a Hyperbola

Illustration showing how the a hyperbola is symmetrical with respect to its conjugate axis.

Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.

Hyperbola Conjugate Diameters

Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.

Illustration showing how a hyperbola con be constructed by points, having been given the foci and the constant difference 2a.

Construction of Hyperbola

Illustration showing how a hyperbola con be constructed by points, having been given the foci and the…

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis BC, vertexes Aa, and foci ff'. Set off any desired number of parts on the axis below the focus, and number them 1,2,3,4,,5,etc. Take the distance a1 as radius, and, with f' as center, strike the cross 1 with f'1=a1. With the distance A1, and the focus f as center, strike the cross 1 with the radius F1=A1, and the cross 1 is a point in the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis…

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse axis BC, foci f' and f, and the vertexes A and a. Take a rule and fix it to a string at e; fix the other end of the string at the focus f. The length of the string should be such that when the rule R is in the position f'C, the loop of the string should reach to A; then move the rule on the focus f', and a pencil at P, stretching string, will trace the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse…

Diagram showing how to construct a hyperbola when given the two foci and the length of the major axis (2a).

Construction of a Hyperbola

Diagram showing how to construct a hyperbola when given the two foci and the length of the major axis…

Illustration showing the definition of an hyperbola. "An hyperbola may be described by the continuous motion of a point, as follows: To one of the foci F' fasten one end of a rigid bar F'B so that it is capable of turning freely about F' as a center in the plane of the paper."

Demonstration of Hyperbola Definition

Illustration showing the definition of an hyperbola. "An hyperbola may be described by the continuous…

Illustration of a hyperbola with distances to foci drawn. "The difference of the distances of any point from the foci of an hyperbola is greater than or less than 2a, according as the point is on the concave or convex side of the curve."

Foci Distance of Hyperbola

Illustration of a hyperbola with distances to foci drawn. "The difference of the distances of any point…

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola a line is drawn bisecting the angle between the focal radii, every point in this line except P is on the convex side of the curve.

Line Bisecting Angle Between Focal Radii in Hyperbola

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola…

Illustration of a point on a hyperbola. "If d denotes the abscissa (x-coordinate) of a point of an hyperbola, r and r' its focal radii, then r = ed - a, and r' = ed + a."

Point on a Hyperbola

Illustration of a point on a hyperbola. "If d denotes the abscissa (x-coordinate) of a point of an hyperbola,…

Illustration showing how to draw a tangent to an hyperbola from a given point P on the convex side of the hyperbola.

Tangent to Hyperbola

Illustration showing how to draw a tangent to an hyperbola from a given point P on the convex side of…

Diagram part of a hyperbola with a tangent line that illustrates "A line through a point on the hyperbola and bisecting the internal angle between the focal radii is a tangent."

Tangent to a Hyperbola

Diagram part of a hyperbola with a tangent line that illustrates "A line through a point on the hyperbola…

All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.

Hyperbola Tangent Triangles

All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances from two fixed points is the square of half the distance between the points. It is a particular case of the Cassinian oval and resembles a figure 8. When the line joining the two fixed points is the axis of x and the middle point of this line is the origin, the Cartesian equation is the fourth degree equation, (((x^2)+(y^2))^2)=2(a^2)((x^2)-(y^2)). The polar equation is (ℽ^2) = 2(a^2)cos(2θ). The locus of the feet of the perpendiculars from the center of an equilateral hyperbola to its tangents is a lemniscate. The name lemniscate is sometimes given to any crunodal symmetric quartic curve having no infinite branch. The name is also sometimes given to a general class of curves derived from other curves in the way that the above is derived from the equilateral hyperbola. With these more general definitions of the lemniscate the above curve is called the lemniscate of Bernoulli.

Lemniscate

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances…

Illustration showing intersecting straight lines meeting to form a triangle. It is formed by the intersection of the surface of a cone with a plane parallel to the axis and actually containing the axis.

Degenerate Conic Forming Triangle

Illustration showing intersecting straight lines meeting to form a triangle. It is formed by the intersection…