Free · Mathematics models
Axis-Aligned Ellipse Foci
Locate foci and the horizontal and vertical semiaxis endpoints for a translated ellipse.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
An ellipse centered at (1,2) with horizontal semiaxis 5 and vertical semiaxis 3 has foci (−3,2) and (5,2).
Focal offset = sqrt(abs(a²−b²)); it lies along the larger semiaxis.
Worked example
Enter these known values and leave the other values blank.
- Center, x-coordinate (c₁)
- 1
- Center, y-coordinate (c₂)
- 2
- a
- 5
- b
- 3
- x-coordinate
- -3
- y-coordinate
- 2
- x-coordinate
- 5
- y-coordinate
- 2
- x-coordinate
- -4
- y-coordinate
- 2
- x-coordinate
- 6
- y-coordinate
- 2
- x-coordinate
- 1
- y-coordinate
- -1
- x-coordinate
- 1
- y-coordinate
- 5
Assumptions and limitations
- The semiaxes are positive and aligned with coordinate axes. Either horizontal or vertical orientation is supported; equal semiaxes give coincident foci.
- Endpoint controls named vertices follow the captured horizontal/vertical directions even when the vertical semiaxis is larger. Hidden perimeter uses Ramanujan’s approximation.
- Rotated ellipses and measurement uncertainty are omitted.
- Clear the unknown numerical field before using a supported inverse; selection controls retain their captured branches.
Common questions
What assumptions matter for this model?
The semiaxes are positive and aligned with coordinate axes. Either horizontal or vertical orientation is supported; equal semiaxes give coincident foci.
What does this model leave out?
Rotated ellipses and measurement uncertainty are omitted.
References
Model requirements: Horizontal semiaxis must be positive. Vertical semiaxis must be positive. Authored accessible explanations preserve each captured diagram mode and its geometric relationships.