Determining The Origin of an Ellipse
By Kathleen Cantor, 03 Jul 2020
An ellipse is a stretched circle, literally speaking. This is a shape that results from transecting a plane through a conical shape, and as such, both a circle and an ellipse are examples of conical sections.
The conic shape of an ellipse is determined by the angle of intersection between the plane and the cone; otherwise, a parabola or hyperbola can be formed. In essence, a circle is an ellipse with the same radius across its diameter, and ellipses are created by stretching circles along the x or y-axis, in any direction.
How are ellipses formed?
Since an ellipse is formed by intersecting a right circular cone with a plane, its best definition begins with a circle to graphically transform and arrive at an ellipse's general formula. The distance from the center of the cycle to any point is the same and has only one constant radius.
Changing a circle into an ellipse involves stretching or squeezing the radius so that distances between the center and the edges are not constant. Distorting the circular shape means there is no symmetry, and a scaling factor is applied to the coordinate factors.
Conic sections can be described by coordinate points of the planar intersection, and these coefficients form a variable equation that determines the shape of the ellipse. From two fixed points on the ellipse, the sum of the x and y distances is a constant, and each of these points is known as a focus or foci in the plural.
Two types of ellipse shapes can be formed, a vertical major axis ellipse and a horizontal major axis ellipse, with the difference being in their orientation.
Properties of an ellipse
Dependent on the orientation of an ellipse, i.e., whether vertical or horizontal, an equation can be used to determine the association of specific coordinates. When a cone is intersected at its base by a plane, an ellipse is formed, and so are two focal points or foci.
The constant sum of distances from the foci has to be arrived at, and an ellipse also has a center, major and minor axes.
With these associations reversed for the opposite orientation, an ellipse will consist of;
- A center: The center of an ellipse is where both the x and y-axis intersect coordinates k and h are
- Major axis: The longest width of an ellipse, where it stretches longer, is its major axis. Horizontal ellipses have their major axis parallel to their x-axis, while vertical ones will have it on their y-axis.
- Minor axis: This is the shortest width of an ellipse, and for horizontal ellipses, their minor axis is on their y-axis. The minor axis has a length, and its coordinate endpoints are known as minor axis co-vertices.
- The Foci: These are two focal points that characterize an ellipses' curvature and shape, and they have focal length coordinates.
- Eccentricity: Conic sections such as ellipses have eccentricity values, which have range values. At the eccentricity of zero, an ellipse becomes a circle, while an eccentricity of one gives rise to a parabola conic shape.
Examples of ellipses with low eccentricity are the orbit of planets around their stars, and the orbit of moons to planets, which are nearly circular. Comets, on the other hand, have high eccentricity in their orbits around the sun.
Elliptical orbits of planets and comets use the sun as one of their focal points.
Ellipses that are centered at the origin
Having defined horizontal and vertical ellipse orientations as positions based on the coordinate plane, other ellipses exist, including those with rotated coordinates. Horizontal and vertical ellipses have axes that lie on either the y and x-axis, and two cases must be considered to work with them.
- Whether an ellipse is centered at the origin
- If the ellipse is centered at points other than the origin
To bridge the geometric and algebraic relationship representations of ellipses, a standard form of their equation is vital. Features of ellipses can be identified by the equation standard form to categorize ellipses by variations of position and the location of the center.
A mental picture of the ellipse can then be formed by interpreting horizontal, vertical, origin centered, and not origin centered ellipses.
The ellipse standard form equation centered at the origin is x2a2 + y2b2 = 1 given the center is 0, 0, while the major axis is on the x-axis. In this equation;
- 2a is the length of the major axis
- Vertices coordinates are a and 0
- 2b is the length of the minor axis
- Co-vertices coordinates are 0 and b
- Where c2 = a2 – b2, the foci coordinates are c and 0
The foci, vertices, and co-vertices are made to relate through this equation, c² = a² - b². This relationship can be used to find the standard form equation of that ellipse.
Writing the ellipse equation when it’s centered at the origin
Ellipses are asymmetrical, meaning that the vertices' coordinates are centered around its origin, and will always be represented by a,0 and c,0, as the shape taken by the conic section. Foci coordinates have the equation representation c,0 or 0,c.
As such, the a and c from a given point can be used within this equation, c² = a² - b² to arrive at the b² coordinates. Before applying his equation, however, you need to determine which axis, x or y that the major axis of the ellipse lies
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