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 xcoordinate of the center of the ellipse; k is the ycoordinate of the center of the ellipse; a is onehalf the length of the longer axis of the ellipse (the longer of the width or length of the ellipse); b is onehalf the length of the shorter axis of the ellipse (the shorter of the width or length of the ellipse); x is a value of xcoordinate of the given point "P" on the circumference of the ellipse; and y is a value of a ycoordinate 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 xaxis and the yaxis of the coordinate system and that these vertexes are symmetrical about the origin of the center of the ellipse and the origin of the xy coordinate system.
References
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