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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.

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,…

A curved line with endpoints. This illustration resembles a parabola.

Line, Curved

A curved line with endpoints. This illustration resembles a parabola.

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…

Illustration of the rolling of equal parabolas. "The two parabolas may be considered as two ellipses with one focus of each removed to infinity."

Rolling of Equal Parabolas

Illustration of the rolling of equal parabolas. "The two parabolas may be considered as two ellipses…

Also called a parabolic spiral, it is a type of Archimedean Spiral. A spiral is defined as "a plane curve which runs continuously round and round, a fixed point, called the center, with constantly increasing radius vector, so that the latter is never normal to the curve; also, a part of such a curve in the course of which the radius from the center describes 360 degrees." &mdash;Whitney, 1889

Fermat's Spiral

Also called a parabolic spiral, it is a type of Archimedean Spiral. A spiral is defined as "a plane…

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…

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

Parabola

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

Illustration showing a parabola as a curve formed by the intersection of the surface of a cone with a plane parallel to a straight line of the surface of the cone.

Parabola

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

Illustration showing a parabola with curve lines.

Parabola

Illustration showing a parabola with curve lines.

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…

Illustration used to draw a parabola when given the base and height.

Construction of Parabola Given Base and Height

Illustration used to draw a parabola when given the base and height.

Diagram of a cone intersected by plane J to form a parabola. Also pictured is a circle formed by the intersection of plane K with the cone. "Every point of a parabola is equidistant from the focus and the directrix."

Cone Intersected by a Plane to Form a Parabola

Diagram of a cone intersected by plane J to form a parabola. Also pictured is a circle formed by the…

Illustration showing the definition of a parabola as a conic section. "The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola."

Conic Section Showing Parabola

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

Illustration showing the definition of a parabola as a conic section. "The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola."

Conic Section Showing Parabola

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

Illustration showing the definition of a parabola as a conic section. The plane cuts the cone parallel to one element.

Conic Section Showing Parabola

Illustration showing the definition of a parabola as a conic section. The plane cuts the cone parallel…

Illustration showing the definition of a parabola as a conic section. The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola.

Conic Section Showing A Parabola

Illustration showing the definition of a parabola as a conic section. The section of a right circular…

An illustration showing the intersection of a plane and a cone. The cone is intersected by a plane parallel to a side. Thus forming a parabola.

Conic Section Showing A Parabola

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

An illustration showing how to construct a parabola by plotting. "Having given the axis, vertex, and focus of the parabola. Divide the transverse axis into any desired number of parts 1, 2, 3, etc., and draw ordinates through the divisions; take the distance A1, and set it off on the 1st ordinate from the focus f to a, so that A1 = fa. Repeat the same operation with the other ordinates - that is, set off the distance A5 from f to e, so that A5 = fe; and so the parabola is constructed."

Construction Of A Parabola

An illustration showing how to construct a parabola by plotting. "Having given the axis, vertex, and…

An illustration showing how to construct a parabola using a pencil and a string. "Having given the two axes, vertex, and focus of the parabola. Take a square cde, and fix to it a string at c; fix the other end of the string at the focus f. The length of the sting should be such that when the square is in the position of the axis Af, the string should reach to the vertex A. Move the square along BB, and the pencil P will describe the parabola."

Construction Of A Parabola

An illustration showing how to construct a parabola using a pencil and a string. "Having given the two…

An illustration showing how to construct a parabola. "Given the vertex A, axis x, and a point P. Draw AB at right angle to x, and BP parallel to x, divide AB and BP into an equal number of equal parts. From the vertex A draw lines to the divisions on BP, from the divisions on AB draw the ordinates parallel to x, the corresponding intersections are points in the parabola."

Construction Of A Parabola

An illustration showing how to construct a parabola. "Given the vertex A, axis x, and a point P. Draw…

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take A as a center and describe a semicircle from B which gives the focus of the parabola at f. Draw any ordinate y at right angle to the abscissa Ax, take a as radius and the focus f as a center, then intersect the ordinate y, by a circle-arc in P which will be a point in the parabola. In the same manner the whole Parabola is constructed."

Construction Of A Parabola

An illustration showing how to construct a parabola. "Given the axis of ordinate B, and vertex A. Take…

Diagram showing how to construct a parabola when given the directrix (d), the focus (F) and the eccentricity (m).

Construction of a Parabola

Diagram showing how to construct a parabola when given the directrix (d), the focus (F) and the eccentricity…

Illustration of a parabola showing that any point of a parabola is the mean proportional between the latus rectum (focal chord) and the abscissa (x-coordinate).

Parabola With Coordinates and Latus Rectum

Illustration of a parabola showing that any point of a parabola is the mean proportional between the…

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its distance from a fixed point in the plane is always equal to its distance from a fixed line in the plane.

Parabola With Focus and Directrix

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its…

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its distance from a fixed point (focus) in the plane is always equal to its distance from a fixed line (directrix) in the plane.

Parabola With Focus and Directrix

Illustration of a parabola - a curve which is the locus of a point that moves in a plane so that its…

Illustration of a parabola showing point P is equidistant to the focus and the directrix.

Point on Parabola

Illustration of a parabola showing point P is equidistant to the focus and the directrix.

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle between PF and the perpendicular from P to the directrix, every point of the line PT, except P, is without the curve."

Point on Parabola

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle…

Illustration of a tangent line drawn from an external point to a parabola.

Tangent to a Parabola

Illustration of a tangent line drawn from an external point to a parabola.

Illustration showing that the area of a parabolic segment made by a chord is two thirds the area of the triangle formed by the chord and the tangents drawn through the ends of the chord.

Tangents and Chords of a Parabola

Illustration showing that the area of a parabolic segment made by a chord is two thirds the area of…

Illustration showing that if two tangents RP and RQ are drawn from a point R to a parabola, the line drawn through R parallel to the axis bisects the chord of contact.

Tangents to a Parabola

Illustration showing that if two tangents RP and RQ are drawn from a point R to a parabola, the line…