Co-Vertices of Ellipse Calculator

Calculator Inputs

Center h

Center k

Semi-major axis a

Minor axis input mode

Semi-minor axis b

Focus distance c

Eccentricity e

Full minor axis length

Ellipse orientation

Rotation angle in degrees

Decimal places

Formula Used

The calculator treats the ellipse center as (h, k). It treats a as the semi-major axis. It finds or accepts b as the semi-minor axis.

For a horizontal ellipse, the co-vertices are (h, k + b) and (h, k - b).

For a vertical ellipse, the co-vertices are (h + b, k) and (h - b, k).

For a rotated ellipse, use angle θ from the positive x-axis. The co-vertices are (h - b sinθ, k + b cosθ) and (h + b sinθ, k - b cosθ).

When focus distance is entered, the calculator uses b = √(a² - c²). When eccentricity is entered, it uses b = a√(1 - e²).

How to Use This Calculator

Enter the center point values h and k.
Enter the semi-major axis value a.
Select how the semi-minor axis should be found.
Choose horizontal, vertical, or rotated orientation.
Enter a rotation angle only for a rotated ellipse.
Choose the number of decimal places.
Press Calculate to show results below the header.
Use CSV or PDF buttons when you need a file.

Example Data Table

Case	Center	a	b or input	Orientation	Co-vertices
Horizontal	(0, 0)	5	b = 3	0 degrees	(0, 3), (0, -3)
Vertical	(2, 1)	6	b = 2	90 degrees	(0, 1), (4, 1)
From focus	(0, 0)	5	c = 4	0 degrees	(0, 3), (0, -3)
Rotated	(1, 2)	8	b = 4	30 degrees	(-1, 5.4641), (3, -1.4641)

Understanding Ellipse Co-Vertices

Co-vertices are the two end points of the minor axis. They sit across the narrow side of an ellipse. The center is exactly halfway between them. These points help describe the shape quickly. They also help when sketching, checking equations, or preparing coordinate data.

Why This Calculator Helps

Manual work can be easy for a level ellipse. It becomes slower when the ellipse is vertical or rotated. This calculator handles all three cases. You can enter a direct semi-minor value. You can also derive it from focus distance or eccentricity. The tool then returns both co-vertices, the minor span, the major vertices, foci, area, and eccentricity.

Practical Uses

Students can use the result to verify graphing homework. Designers can use it when plotting oval paths. Developers can use the exported file for reports or testing. Survey style layouts can also benefit from fast coordinate output. The calculator keeps the process clear. It shows each important value in one result panel.

Input Choices

The center values set the anchor point. The semi-major axis defines the long radius. The minor input mode controls how the short radius is found. Direct mode uses the semi-minor axis. Focus mode uses the relationship between a, b, and c. Eccentricity mode uses a ratio. Minor length mode divides the full minor axis by two.

Rotation Support

A horizontal ellipse uses an angle of zero degrees. A vertical ellipse uses ninety degrees. A rotated ellipse uses the custom angle you enter. The angle is measured from the positive x-axis toward the major axis. The calculator builds the minor axis direction from that angle. It then places each co-vertex one semi-minor length from the center.

Good Checking Habits

Always keep the semi-major axis at least as large as the semi-minor axis. Check units before exporting. Use the same unit for all lengths. Review the plotted meaning of the angle. A small input mistake can move both co-vertices. The example table gives sample cases for comparison. Use it before entering real project values.

Output Review

Use the rounded values for presentation. Use the raw CSV values for deeper checks. Compare both co-vertices with the center. Their midpoint should match the center, even after rotation each time.

FAQs

What are co-vertices of an ellipse?

Co-vertices are the endpoints of the minor axis. They show the narrow width of the ellipse and lie equally away from the center.

What inputs are required?

You need the center, semi-major axis, and enough minor axis data. You may enter b directly or derive it from focus distance, eccentricity, or full minor length.

Can this calculator handle a rotated ellipse?

Yes. Select rotated orientation and enter the major axis angle in degrees. The calculator then returns rotated co-vertex coordinates.

What happens in focus distance mode?

The calculator uses the relationship b equals the square root of a squared minus c squared. The focus distance must be less than a.

What is eccentricity mode?

Eccentricity mode finds b from a and e. The value of e must be at least zero and less than one.

Why must a be larger than b?

The semi-major axis is the long radius of the ellipse. The semi-minor axis cannot be longer than that value.

What does the CSV file include?

The CSV file includes the center, axes, co-vertices, vertices, foci, eccentricity, area, orientation, and angle.

Is the PDF export useful for reports?

Yes. The PDF export gives a compact result summary. It is useful for records, homework notes, and quick project documentation.