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"When the minor axis is at least two-thirds the major, the following method ma be used: Make CF and CG equal to AB-DE. Make CH and CI equal to 3/4 CF F, G, H, I will be centers for arcs E, D, B, and A." —French 1911

Approximate Ellipse using Lines

"When the minor axis is at least two-thirds the major, the following method ma be used: Make CF and…

"This (five centered arc) method is based on the principle that the radius of curvature at the end of the minor axis is the third proportional to the semi-minor and semi-major axes, and similarly at the end of the major axis is the third proportional to the semi-major and semi-minor axes. The intermediate radius found is the mean proportional between these two radii." —French, 1911

Approximate Ellipse with Five Centered Arc

"This (five centered arc) method is based on the principle that the radius of curvature at the end of…

"Join A and D. Lay off DF equal to AC-DC. Bisect AF by a perpendicular crossing AC at G and intersecting DE produced at H. Make CG' equal to CG and CH' equal to CH. Then G, G', H, and H' will be centers for four arcs approximating the ellipse. The half of this ellipse when used in masonry construction is known as the three-centered arch." —French, 1911

Approximate Ellipse with Four Centers

"Join A and D. Lay off DF equal to AC-DC. Bisect AF by a perpendicular crossing AC at G and intersecting…

Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle.

Circular Involution

Every elliptical involution has the property that there are two definite points in the plane from which…

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

Cone

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

The lines which join corresponding points in an involution on a conic all pass through a fixed point; and reciprocally, the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line.

Conic Involution

The lines which join corresponding points in an involution on a conic all pass through a fixed point;…

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…

An ellipsograph uses the trammel method to create ellipses.

Construction of Ellipse using Ellipsograph

An ellipsograph uses the trammel method to create ellipses.

Create diameters with major, minor axes, or a pair of conjugate diameters. Construct a parallelogram, then divide into equal parts creating intersections for the curve.

Construction of Ellipse using Parallelogram

Create diameters with major, minor axes, or a pair of conjugate diameters. Construct a parallelogram,…

An ellipse can be constructed by pinning three pins and tying an inelastic in tightly around the pins.

Construction of Ellipse using Pin and String

An ellipse can be constructed by pinning three pins and tying an inelastic in tightly around the pins.

Trammel method is a construction of an ellipse using a strip of paper, thin cardboard or sheet celluloid.

Construction of Ellipse using Trammel Method

Trammel method is a construction of an ellipse using a strip of paper, thin cardboard or sheet celluloid.

"Any noncircular curve may be approximated by tangent circle arcs, selecting a center by trial, drawing as much of an arc as will practically coincide with the curve, then changing the center and radius for the next portion, remembering always that if arcs are to be tangent, their centers must lie on the common normal at the point of tangency." —French 1911

Curve Inked with Circle Arcs

"Any noncircular curve may be approximated by tangent circle arcs, selecting a center by trial, drawing…

Illustration of a cylinder cut by a plane making an angle of 57° with the base.

Plane Intersecting A Cylinder

Illustration of a cylinder cut by a plane making an angle of 57° with the base.

Illustration of the method of finding the projection, in the form of an ellipse, of the top of a cylinder greatly inclined to a plane.

Projection of Cylinder

Illustration of the method of finding the projection, in the form of an ellipse, of the top of a cylinder…

Using C as a center, draw two circles with different diameters. The intersection of the diameter lines will determine the points on the curve.

Determining Points on Ellipse Using Circles

Using C as a center, draw two circles with different diameters. The intersection of the diameter lines…

"In geometry, an angle connected with an ellipse and defined as ... angle BCL, reckoned from one determinate end, B, of the transverse axis, called the eccentric angle of the point H." -Whitney, 1911

Eccentric Angle

"In geometry, an angle connected with an ellipse and defined as ... angle BCL, reckoned from one determinate…

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

Ellipse

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

Illustration of an ellipse with major and minor axes, foci, and points on the ellipse. "An ellipse is a plane figure bounded by a curved line, to any point of which the sum of the distances from two fixed points within, called the foci, is equal to the sum of the distances from the foci to any other point on the curve."

Ellipse

Illustration of an ellipse with major and minor axes, foci, and points on the ellipse. "An ellipse is…

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration of an ellipse with major and minor axes.

Ellipse With Axes

Illustration of an ellipse with major and minor axes.

Illustration of an ellipse.

Ellipse

Illustration of an ellipse.

Illustration showing an ellipse formed by the intersection of the surface of a cone with a plane neither parallel nor perpendicular to a straight line of the surface of the cone nor the axis.

Ellipse

Illustration showing an ellipse formed by the intersection of the surface of a cone with a plane neither…

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its auxiliary circle are in the ratio b:a."

Corresponding Points in an Ellipse and Circle

Illustration of half of an ellipse. "The ordinates of two corresponding points in an ellipse and its…

Illustration used to draw a an ellipse using string and pins by describing a circles with diameters equal to the minor and major axes of the ellipse.

Construction of Ellipse by Describing Circles

Illustration used to draw a an ellipse using string and pins by describing a circles with diameters…

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a semicircle on AB, draw ordinates in the circle at right angle to AB, the corresponding and equal ordinates for the ellipse to be drawn parallel to the lines, and thus the elliptic curve is obtained as shown by the figure."

Construction Of An Ellipse Tangent To Two Parallel Lines

An illustration showing how to construct an ellipse parallel to two parallel lines A and B. "Draw a…

Illustration used to draw a an ellipse using string and pins.

Construction of Ellipse Using String

Illustration used to draw a an ellipse using string and pins.

Illustration of a ellipse with squares drawn to demonstrate area.

Area of Ellipse

Illustration of a ellipse with squares drawn to demonstrate area.

Illustration of half of an ellipse used to show that the area of an ellipse is equal to pi times a times b.

Area of an Ellipse

Illustration of half of an ellipse used to show that the area of an ellipse is equal to pi times a times…

Illustration showing the definition of an ellipse as a conic section. "The section of a right circular cone made by a plane that cuts all the elements of the surface of the cone is an ellipse."

Conic Section Showing an Ellipse

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

Illustration showing the definition of an ellipse as a conic section. "The section of a right circular cone made by a plane that cuts all the elements of the surface of the cone is an ellipse."

Conic Section Showing an Ellipse

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

Illustration showing the definition of an ellipse as a conic section. The plane cuts the cone obliquely.

Conic Section Showing Ellipse

Illustration showing the definition of an ellipse as a conic section. The plane cuts the cone obliquely.

Illustration showing the definition of an ellipse as a conic section. The section of a right circular cone made by a plane that cuts all the elements of the surface of the cone is an ellipse. The plane is neither parallel nor perpendicular to the axis.

Conic Section Showing An Ellipse

Illustration showing the definition of an ellipse 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 neither parallel to nor perpendicular to the base, nor parallel to a side. Thus forming an ellipse.

Conic Section Showing An Ellipse

An illustration showing the intersection of a plane and a cone. The cone is intersected by a plane neither…

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Conjugate Diameters of an Ellipse

Illustration showing that if one diameter is conjugate to a second, the second is conjugate to the first.

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points, having given its two axes.

Construction of an Ellipse

Illustration of half of an ellipse and its auxiliary circle used to construct an ellipse by points,…

An illustration showing how to construct an ellipse. "With a as a center, draw two concentric circles with diameters equal to the long and short axes of the desired ellipse. Draw from o any number of radii, A, B, etc. Draw a line Bb' parallel to n and bb' parallel to m, then b is a point in the desired ellipse.

Construction Of An Ellipse

An illustration showing how to construct an ellipse. "With a as a center, draw two concentric circles…

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set off from c half the great axis at a and b, which are the two focuses of the ellipse. Take an endless string as long as the three sides in the triangle abc, fix two pins or nails in the focuses, one in a and one in b, lay the string around a and b, stretch it with a pencil d, which then will describe the desired ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set…

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three equal parts, draw the two circles, and where they intersect one another are the centers for the tangent arcs of the ellipses as shown by the figure."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Divide the long axis into three…

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off the short axis from A to b, divide b into three equal parts, set off two of these parts from o towards c and c which are the centers for the ends of the ellipse. Make equilateral triangles on cc, when ee will be the centers for the sides of the ellipse. If the long axis is more than twice the short one, this construction will not make a good ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using circle arcs. "Given the two axes, set off…

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis from c to ff, which will be the two focuses in the ellipse. Divide the long axis into any number of parts, say a to be a division point. Take Aa as radius and f as center and describe a circle arc about b, take aB as radius and f as center describe another circle arc about b, then the intersection b is a point in the ellipse, and so the whole ellipse can be constructed."

Construction Of An Ellipse

An illustration showing how to construct an ellipse. Given the two axes, set off half the long axis…

Illustration used to draw a an ellipse with major axis AB and minor axis CD.

Construction of Ellipse

Illustration used to draw a an ellipse with major axis AB and minor axis CD.

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

Construction of an Ellipse

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

Diagram of an ellipse that can used to illustrate the definition. "The constant ration between the distances of a point on an ellipse from the focus and the directrix equals the linear eccentricity divided by the semi major axis."

Definition of Ellipse

Diagram of an ellipse that can used to illustrate the definition. "The constant ration between the distances…

Illustration showing the definition of an ellipse. "An ellipse is a curve which is the locus of a point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant." The foci and major axis is drawn.

Demonstration of Ellipse Definition

Illustration showing the definition of an ellipse. "An ellipse is a curve which is the locus of a point…

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its focal radii, then r'=a+ed, r=a-ed."

Focal Radii of an Ellipse

Illustration of half of an ellipse. "If d denotes the abscissa of a point of an ellipse, r and r' its…

Diagram that illustrates "The algebraic sum of the distances of any point from the foci of a conic is greater or less than 2a, according as the point is without or within the curve."

Foci Distances on an Ellipse

Diagram that illustrates "The algebraic sum of the distances of any point from the foci of a conic is…

A graph of tan ellipse with Foci and Center.

Graph of Ellipse with Foci and Center Labeled

A graph of tan ellipse with Foci and Center.

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal to the length of the major axis of the ellipse. The ellipse is externally tangent to the circle.

Ellipse Inscribed In A Circle

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal…

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Ellipse With Parts Labeled

Illustration of an ellipse with foci F' and F, major axis A' to A, minor axis B' to B, and center O.

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the angle between one of the focal radii and the other produced, every point in this line except P is without the curve."

Line Bisecting Angle Between Focal Radii on Ellipse

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the…

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product of the segments of the major axis made by the ordinate as the square of b to the square of a.

Ordinate and Major Axis of Ellipse

Illustration of half of an ellipse. The square of the ordinate of a point in an ellipse is to the product…

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Parallel Tangents to an Ellipse

Illustration showing that tangents drawn at the ends of any diameter are parallel to each other.

Illustration of ellipse with parts labeled.

Ellipse With Parts Labeled

Illustration of ellipse with parts labeled.