ClipArt ETC

This mathematics ClipArt gallery offers 41 images of conic sections, or conics, in the shape of an ellipse. Conics are obtained by taking a cone, or conical surface, and intersecting it with a plane. Conic ellipses are retrieved when the intersecting cone and plane create a closed curve, except for the case where the plane that creates the closed curve is parallel to the plane of the circle created by the cone. In this case, a conic circle is created.

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;…

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 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…

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 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 of a cone cut by a plane that is not parallel to the base and does not intersect the base.

Conic Section Ellipse

Illustration of a cone cut by a plane that is not parallel to the base and does not intersect the base.

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.

Illustrations used to construct an ellipse. Fig. 59 shows a compass at C used to strike a circle. Where the circle intersects the line at D and B, place pins. These pins represent the foci of the ellipse. Use a cord and a pencil to draw the ellipse as shown in figure 60. This illustrates the definition of an ellipse.

Construction Of An Ellipse

Illustrations used to construct an ellipse. Fig. 59 shows a compass at C used to strike a circle. Where…

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…

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.

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis, segment CD is the conjugate (minor) axis, and point O is the center of the ellipse. Both foci are also labeled in the illustration.

Parts of Ellipse

Diagram of an ellipse that can used to illustrate the different parts. Segment MN is the major axis,…

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse is greater than or less than 2a, according as the point is without or within the curve."

Point Distances to Foci on Ellipse

Illustration of half of an ellipse. "The sum of the distances of any point from the foci of an ellipse…

Illustration of how to draw a tangent to an ellipse from an external point.

Tangent From External Point to an Ellipse

Illustration of how to draw a tangent to an ellipse from an external point.

Diagram an ellipse with a tangent line that illustrates "A line through a point on the ellipse and bisecting the external angle between the focal radii is a tangent."

Tangent to an Ellipse

Diagram an ellipse with a tangent line that illustrates "A line through a point on the ellipse and bisecting…

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary circle cut the major axis produced at the same point.

Tangents to an Ellipse

Illustration showing the tangents drawn at two corresponding points of an ellipse and its auxiliary…

"If two equal ellipses, each turning about one of its foci, are placed in contact in such a way that the distance between the axes O<SUB>1</SUB> and O<SUB>2</SUB> is equal to the major axis of the ellipses, we shall find that they will be in contact on the line of centres and that the rolling arcs are of equal length."

Rolling of Equal Ellipses

"If two equal ellipses, each turning about one of its foci, are placed in contact in such a way that…

The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point. This is an ellipse.

Conic Foci Involution

The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal…

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide OA and OB each into three equal parts; draw the quadrant AC. From C, draw the line Cc through the point 1. Through the points 2 draw de at an angle of 45&deg; with the major axis. Then 2 is the center for the ends of the ellipse; e is the center for the arc dc; and C is the center for the arc cf."

Construction Of An Isometric Ellipse

An illustration showing how to construct an isometric ellipse by compass and six circle arcs. "Divide…