Name: ellipse geometry: foci, axes, and equations
Uploaded: 2025-11-26T05:44:08.000Z
Duration: 1 min 30 s
Channel: JoVE

An ellipse forms when a right circular cone is sliced by an angled plane that doesn&#8217;t intersect its base, creating a closed curve.Geometrically, an ellipse is the set of all points for which the sum of the distances to two fixed points&#8212;called foci&#8212;is constant.The longest diameter is the major axis, and the shortest is the minor axis. These axes intersect at the center, and the major axis ends at the vertices.The standard form is derived by centering the origin and placing the foci on the x-axis at -c and +c, where c is the distance from the center to each focus.From any point on the ellipse, the total distance to both foci is the length of the major axis, expressed as a sum of two square roots. Rearranging and squaring both sides eliminates one square root. Squaring again removes all square roots, producing an equation with squared terms. Substituting the relationship between the axis lengths and focal distance, then rearranging, gives the standard form.The larger denominator corresponds to the major axis, regardless of the ellipse&#8217;s center.This equation describes real-world elliptical paths like planetary orbits and satellite motion.

ellipses

Ellipses

An ellipse is formed when a right circular cone is intersected by an inclined plane that does not cut through its base. This intersection yields a closed, symmetric curve characterized by distinctive geometric properties. Most notably, an ellipse is defined as the collection of all points in a plane for which the combined distances to two fixed points&#8212;called the foci&#8212;remain constant.The ellipse features two principal axes: the major and the minor axes. The major axis is the longest diameter, passing through both foci and the center of the ellipse. The minor axis is the shortest diameter and is oriented perpendicular to the major axis at the ellipse's center. For any point on the ellipse, the sum of the distances to the two foci is equal to the length of the major axis.The general equation of an ellipse with its center at the origin (0,0) is:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/189;' src='https://cloudfront.jove.com/files/ftp_upload/40088/40088_Equation_1_editor_40088.svg' alt='40088_Equation_1.svg' /></figure>Here, a and b represent the lengths of the semi-major and semi-minor axes, respectively, with a &gt; b. If the center is at a point (h, k), the equation shifts to:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/189;' src='https://cloudfront.jove.com/files/ftp_upload/40088/40088_Equation_2_editor_40088.svg' alt='40088_Equation_2.svg' /></figure>This canonical form is essential in modeling real-world elliptical trajectories, including the elliptical orbits described by Kepler's laws. It illustrates the deep connection between geometry and physics.

An ellipse is formed when a right circular cone is intersected by an inclined plane that does not cut through its base. This intersection yields a closed, ...

Education

JoVE Core

Math Fundamentals

Unit

Analytic Geometry

KEY TERMS AND CONCEPTS

ellipse geometry: foci, axes, and equations

An ellipse is a closed curve in geometry with two foci, a major axis, and a minor axis. It is formed when a right circular cone is cut by an inclined plane that does not pass through the cone’s base. The shape is symmetric, which makes its structure easy to study in analytic geometry.

One key way to define an ellipse is by distance. For any point on the curve, the sum of the distances to the two fixed points, called the foci, stays constant. That constant sum is equal to the length of the major axis. The major axis is the longest diameter and passes through both foci and the center of the ellipse.

The minor axis is the shortest diameter. It is perpendicular to the major axis at the center of the ellipse. In a standard ellipse, the semi-major axis is labeled a and the semi-minor axis is labeled b, with a greater than b. When the center is at the origin, the ellipse has a standard equation. If the center moves to a point (h, k), the equation shifts to a new canonical form.

This equation is useful for modeling real-world paths that follow an elliptical shape. A common example is the elliptical orbit described by Kepler’s laws. The ellipse shows how geometry connects with physics in a direct and useful way.

Ellipse Geometry: Foci, Axes, and Equations

inclination of a line

Slope Angle and Line Direction

The slope angle shows how a line tilts on the Cartesian plane. It gives the line's direction by linking geometry with slope, which tells how steeply the line rises or falls. A line has no thickness, but its orientation still matters. This angle is measured counterclockwise from the positive x-axis. For nonhorizontal lines, the inclination angle ranges from 0 to π radians. That range helps describe the line's tilt in a clear geometric way. When two lines intersect, each line has its own...

parabolas

Parabolas: Focus, Directrix, and Symmetry

A parabola is a conic section with a clear geometric rule. It forms when a plane cuts a double-napped cone parallel to the cone’s slant height. The result is an open curve used in coordinate geometry and defined by a fixed point and a fixed line. That fixed point is the focus, and the fixed line is the directrix. Every point on a parabola stays the same distance from the focus as from the directrix. In mathematics, this set of points is called the locus of points with equal distance from both.<br...

reflective property of parabolas

Parabola Focus and Signal Reflection

A parabola is a U-shaped conic section with an important reflective property. It forms when a plane cuts a double-napped cone in a direction parallel to one side of the cone. In geometry, a parabola is also defined as the set of points that are equally far from a fixed point, called the focus, and a fixed line, called the directrix. This curve has a useful way of handling light, sound, and other parallel rays. Incoming rays that travel parallel to the axis of symmetry reflect toward one point.

An ellipse is a closed curve in geometry with two foci, a major axis, and a minor axis. It is formed when a right circular cone is cut by an inclined plane that does not pass through the cone’s base. The shape is symmetric, which makes its structure easy to study in analytic geometry. One key way to define an ellipse is by distance. For any point on the curve, the sum of the distances to the two fixed points, called the foci, stays constant. That constant sum is equal to the length of the...

eccentricity of an ellipse

Ellipse Shape and Orbits Explained

An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse. The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When...

hyperbolas

Hyperbolas: Foci, Vertices, and Asymptotes

Hyperbolas are conic sections with two separate branches. They form when a plane cuts through a double-napped cone at a steeper angle than the cone’s slope, so the plane passes through both nappes. The two curves are mirror images and open away from each other along the transverse axis. The closest points on the branches to the center are called vertices. The distance from the center to a vertex is labeled a. The axis perpendicular to the transverse axis is the conjugate axis, and it is linked...

geometry of hyperbolas

Hyperbola Properties and Asymptotes

A hyperbola is a curve defined by two fixed points called foci. For any point on the curve, the absolute difference in its distances to the foci stays constant. This distance rule gives the hyperbola its shape and separates it from other conic sections. The standard equation shows two branches that extend forever. Each branch moves closer to two asymptotes, which are lines that guide the curve as it grows. The parameters a and b control key features of the graph. The value a measures the...

polar equations of conics

Conic Sections in Polar Form

Conic sections can be written in polar form using a focus and a directrix. The focus is a fixed point, and the directrix is a fixed line. A point on the conic is defined by the ratio of its distance to the focus and its distance to the directrix. That ratio is called the eccentricity. This setup gives one framework for ellipses, parabolas, and hyperbolas. When the focus is placed at the origin of the polar coordinate system, one polar equation can represent any conic section. That makes the...

key terms and concepts

Learn how an ellipse is formed, how its foci and axes work, and how the standard equation changes when the center moves in analytic geometry.

Ellipse Geometry: Foci, Axes, and Equations | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Ellipse Geometry: Foci, Axes, and Equations

Description

Transcript

Tags