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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…

The cone is sliced by a hyperbola within a plane. The angle of the hyperbola will make a smaller angle than the other elements.

Conic Section Using Hyperbola

The cone is sliced by a hyperbola within a plane. The angle of the hyperbola will make a smaller angle…

Conic sections, cones divided by a plane.

Conic Sections

Conic sections, cones divided by a plane.

Diagram of a cone with spheres and cut by a plane to depict the conic sections.

Cone depicting Conic Sections

Diagram of a cone with spheres and cut by a plane to depict the conic sections.

Diagram of a cone with inscribed spheres and cut by various planes to depict the conic sections: circle, ellipse.

Cone depicting Conic Sections

Diagram of a cone with inscribed spheres and cut by various planes to depict the conic sections: circle,…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. If the focus is outside the circle, we get a hyperbola. If it's inside the circle, we get an ellipse. If the auxiliary circle is a straight line (radius is infinite), we get a parabola.

Construction of a Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. If the focus…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. The focus is on the left of the auxiliary circle, thus producing a very obtuse hyperbola.

Focus In Auxiliary Circle of Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. The focus is…

Diagram showing how to construct a conic when given the focus and the auxiliary circle. As the focus moves inside the circle the ellipse broadens out until the focus reaches the center and becomes a circle.

Focus In Auxiliary Circle of Conic

Diagram showing how to construct a conic when given the focus and the auxiliary circle. As the focus…

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

Hyperbola

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

Generating a hyperbola from two equal and parallel circular disks.

Generate Hyperbola

Generating a hyperbola from two equal and parallel circular disks.

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 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…

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.

Three dimensional representation of a variable hyperbola moving parallel to itself along the parabolas as directrices.

Hyperbolic Parabaloid

Three dimensional representation of a variable hyperbola moving parallel to itself along the parabolas…

Two dimensional representation of variable hyperbola moving parallel to itself along the parabolas as directrices.

Hyperbolic Parabaloid

Two dimensional representation of variable hyperbola moving parallel to itself along the parabolas as…

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…

The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles. the ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.

Parabola Foci Properties

The segments between the point of intersection of two tangents to a conic and their points of contact…

These are skew bevel wheels, made of a hyperboloid of revolution around an axis.

Sliding Contact

These are skew bevel wheels, made of a hyperboloid of revolution around an axis.

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…