What is an ellipse? A geometric figure?

One Ellipse is a flat geometric figure obtained by the intersection between a flat it is a cone. That's why this figure is called conical, just like the circumference, a parable and the hyperbole. The following figure is an example of an ellipse and demonstrates the difference between the geometric representation of this figure and the circumference.

In the figure above, the F points₁ and F₂ they are focusesgivesEllipse, and the distance between them is defined as 2c.

Formal definition of the ellipse

Given the F points₁ and F₂, with the distance 2c between them, the Ellipse it's the setFrompoints P where the following equality is valid:

d_PF1 + d_PF2 = 2nd

In other words, the Ellipse is the set of points in which the sumof thedistances even each of focuses is equal to constant 2a. Thus, we can say that P is a point belonging to an ellipse if the sum of the distances from P to each of the foci is equal to 2a.

The following image illustrates this definition. Note that the sumof thedistances between P and the

focuses gives Ellipse is equal to the sum of the distances from point Q to the focus of the ellipse. Therefore, P and Q belong to this ellipse.

Note that length 2a is always greater than length 2c.

Ellipse Elements

Below, check out a list of the main elementsgivesEllipse and a brief definition of each of them.

Spotlights: in the images in this article, the focuses are the F points₁ and F₂. These are key points at which distances must be evaluated to know whether a point belongs or does not belong to the ellipse.

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center: given the F focuses₁ and F₂, the center of the ellipse is the midpoint of the segment F₁F₂ whose ends are the foci.

Axlebigger: in the image below, the major axis is segment A₁THE₂. Their endpoints are points that belong to the intersection between the ellipse and the line containing the foci. The measure of this axis is equal to 2a, the same length as the sum of the distances between any point on the ellipse and its foci.

Axlesmaller: in the image below, the minor axis is segment B₁B₂. Their endpoints are points that belong to the intersection between the ellipse and the straight line perpendicular to the major axis. The length of this axis is equal to 2b, where b is the distance between the center of the ellipse and point B₁.

Distancefocal: Distance between ellipse foci and is always equal to 2c.

Eccentricity: is the following reason:

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The

The following image illustrates some of the elements of the Ellipse and the lengths representing measures "a", "b" and "c", in which the relationship of Pythagoras: a² = b² + c².

Reduced Ellipse Equations

The first equation reduced of the ellipse is used in the case where the focuses of this figure are on the x-axis and the center of the Ellipse is about the origin of the Cartesian plane:

x² + y² = 1
The² B²

The second equationreduced gives Ellipse is used in the case where the foci of this figure are on the y axis and the center is on the origin of the Cartesian plane:

y² + x²= 1
The² B²

By Luiz Paulo Moreira
Graduated in Mathematics

Would you like to reference this text in a school or academic work? Look:

SILVA, Luiz Paulo Moreira. "What is an ellipse? A geometric figure?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-elipse.htm. Accessed on June 27, 2021.