How to Find the Vertices of an Ellipse

Visually locate the vertices of an ellipse.
••• Jupiterimages/Photos.com/Getty Images

The vertices of an ellipse, the points where the axes of the ellipse intersect its circumference, must often be found in engineering and geometry problems. Computer programmers also must know how to find the vertices to program graphic shapes. In sewing, finding the vertices of the ellipse can be helpful for designing elliptic cutouts. You can find the vertices of an ellipse in two ways: by graphing an ellipse on paper or through the equation of the ellipse.

Graphical Method

    Circumscribe a rectangle with your pencil and ruler such that the midpoint of each edge of the rectangle touches a point on the circumference of the ellipse.

    Label the point where the right rectangle edge intersects the circumference of the ellipse as point "V1" to indicate that this point is the first vertex of the ellipse.

    Label the point where the top rectangle edge intersects the circumference of the ellipse as point "V2" to indicate that this point is the second vertex of the ellipse.

    Label the point where the left edge of the rectangle intersects the circumference of the ellipse as point "V3" to indicate that this point is the third vertex of the ellipse.

    Label the point where the lower edge of the rectangle intersects the circumference of the ellipse as point "V4" to indicate that this point is the fourth vertex of the ellipse.

Finding the Vertices Mathematically

    Find the vertices of an ellipse defined mathematically. Use the following ellipse equation as an example:

    x^2/4 + y^2/1 = 1

    Equate the given ellipse equation, x^2/4 + y^2/1 = 1, with the general equation of an ellipse:

    (x - h)^2/a^2 + (y - k)^2/b^2 = 1

    By doing so, you will obtain the following equation:

    x^2/4 + y^2/1 = (x - h)^2/a^2 + (y - k)^2/b^2

    Equate (x - h)^2 = x^2 to calculate that h = 0 Equate (y - k)^2 = y^2 to calculate that k = 0 Equate a^2 = 4 to calculate that a = 2 and -2 Equate b^2 = 1 to calculate that b = 1 and -1

    Note that for the general equation of the ellipse, h is the x-coordinate of the center of the ellipse; k is the y-coordinate of the center of the ellipse; a is one-half the length of the longer axis of the ellipse (the longer of the width or length of the ellipse); b is one-half the length of the shorter axis of the ellipse (the shorter of the width or length of the ellipse); x is a value of x-coordinate of the given point "P" on the circumference of the ellipse; and y is a value of a y-coordinate of the given point "P" on the circumference of the ellipse.

    Use the following "vertex equations" to find the vertices of an ellipse:

    Vertex 1: (XV1, YV1) = (a - h, h) Vertex 2: (XV2, YV2) = (h - a, h) Vertex 3: (XV3, YV3) = (k, b - k) Vertex 4: (XV4, YV4) = (k, k - b)

    Substitute the values of a, b, h and k (a = 2, a = -2, b = 1, b = -1, h = 0, k = 0) previously calculated to obtain the following:

    XV1, YV1 = (2 - 0, 0) = (2, 0) XV2, YV2 = (0 - 2, 0) = (-2, 0) XV3, YV3 = (0, 1 - 0) = (0, 1) XV4, YV4 = (0, 0 - 1) = (0, -1)

    Conclude that the four vertices of this ellipse are on the x-axis and the y-axis of the coordinate system and that these vertexes are symmetrical about the origin of the center of the ellipse and the origin of the x-y coordinate system.

Related Articles

How to Calculate CG
How to Calculate Ellipse Eccentricity
How to Calculate Angles Without a Protractor
How to Find the Area of a Parallelogram With Vertices
How to Solve a Hexagon
How to Find the Slope in a Circle
How to Calculate the Lineal Footage of a Circle
How to Calculate the Radius and Diameter of an Oval
How to Construct a Rhombus With a Compass & Straight...
What Are Vertices in Math?
How to Calculate the Area of an Oval
How to Find Y Value for the Slope of a Line
How to Calculate the Base of a Shape
How to Calculate Perimeter and Area Ratio
How to Create Linear Equations
How to Convert XY Coordinates to Longitude and Latitude
How to Calculate Eccentricity
How to Convert Diameter to Square Centimeters
How to Calculate Vertical Exaggeration
How to Calculate the Radius From the Circumference

Dont Go!

We Have More Great Sciencing Articles!