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Ellipse Standard Form — result sheet
Construct the standard axis-aligned ellipse equation from a center and positive horizontal and vertical semiaxes.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where a remains the horizontal semiaxis and b remains the vertical semiaxis.
Translate the origin to the center, divide the squared horizontal displacement by a^2 and the squared vertical displacement by b^2, and set their sum to one. The axes are not reordered by length, so horizontal and vertical meaning is preserved.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Center h — coordinate units; The x-coordinate of the ellipse center.; minimum -1000000; maximum 1000000
- Center k — coordinate units; The y-coordinate of the ellipse center.; minimum -1000000; maximum 1000000
- Horizontal semiaxis a — length units; Positive horizontal distance from center to the ellipse boundary.; minimum 1.0E-6; maximum 1000000
- Vertical semiaxis b — length units; Positive vertical distance from center to the ellipse boundary.; minimum 1.0E-6; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Center h | 2 |
| Center k | -3 |
| Horizontal semiaxis a | 5 |
| Vertical semiaxis b | 2 |
The standard equation is (x-2)^2/25 + (y+3)^2/4 = 1, with horizontal semiaxis 5 and vertical semiaxis 2.
Assumptions and limits
- The ellipse is axis-aligned in a flat two-dimensional Cartesian coordinate system.
- a and b are positive semiaxes tied to horizontal and vertical directions respectively; a need not be greater than b.
- The equation is returned as a standard relation, not as an area, circumference, rotated-ellipse fit, or point-membership classification.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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