Ellipse Calculator
Solve horizontal and vertical ellipses for standard form, center $(h, k)$, semi-major axis ($a$), semi-minor axis ($b$), foci coordinates, eccentricity ($e = c/a$), area, and perimeter with interactive SVG plots.
Ellipse Definition & Standard Formulas
An ellipse is the conic section with eccentricity 0 ≤ e < 1, defined as the locus of points where the sum of distances to two foci is constant (2a). The larger denominator under x² or y² determines the orientation of the major axis.
Ellipse Definition & Two-Foci Distance Sum Locus
An ellipse is the geometric locus of all points $P(x, y)$ in a plane such that the sum of its distances from two fixed points $F_1$ and $F_2$ (the foci) is constant and equal to the major axis length $2a$:
The line segment of length $2a$ connecting the vertices through the center and foci is the major axis. The perpendicular line segment of length $2b$ through the center is the minor axis.
Standard Form Equations (Horizontal vs Vertical Major Axis)
- Major axis is horizontal along $y = k$.
- Vertices: $(h \pm a, k)$
- Foci: $(h \pm c, k)$ where $c = \sqrt{a^2 - b^2}$
- Major axis is vertical along $x = h$.
- Vertices: $(h, k \pm a)$
- Foci: $(h, k \pm c)$ where $c = \sqrt{a^2 - b^2}$
Focal Distance ($c = \sqrt{a^2 - b^2}$) & Eccentricity ($e = c/a$)
The Pythagorean relationship $a^2 = b^2 + c^2$ connects semi-major axis $a$, semi-minor axis $b$, and focal parameter $c$:
Distance from center $(h, k)$ along the major axis to each focus point.
Dimensionless ratio $0 \le e < 1$ measuring the degree of elongation / flattening.
Exact Area ($\pi ab$) & Ramanujan Perimeter Approximations
Exact Enclosed Area
Derived by integrating or scaling a unit circle by factor $a$ along x and $b$ along y ($A = \pi \cdot 1^2 \cdot ab$).
Ramanujan Perimeter Formula
Where $h = \frac{(a - b)^2}{(a + b)^2}$. Provides sub-millimeter precision for high-eccentricity ellipses.
Applications in Astronomy (Kepler Orbits) & Acoustics
Kepler's 1st Law (Planetary Orbits)
All planets orbit the Sun in elliptical trajectories with the Sun at one focus $F_1$. Earth's orbital eccentricity is $e \approx 0.0167$.
Acoustic Whispering Galleries
In elliptical domes (e.g. St. Paul's Cathedral), sound waves whispered at focus $F_1$ reflect off the curved ceiling to concentrate at focus $F_2$.
Medical Lithotripsy
Acoustic shockwaves generated at focus $F_1$ inside an elliptical reflector reflect to focus precisely on kidney stones positioned at focus $F_2$.
Graded Step-by-Step Numerical Solutions
Analyze the vertical ellipse $\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{25} = 1$.
1. Center $(h, k) = (2, -1)$.
2. Denominators: $a^2 = 25 \implies a = 5$ (under $y$), $b^2 = 9 \implies b = 3$ (under $x$).
3. Major axis is vertical ($x = 2$). Vertices: $(2, -1 \pm 5) \implies (2, 4)$ and $(2, -6)$.
4. Focal distance $c = \sqrt{25 - 9} = \sqrt{16} = 4$. Foci: $(2, -1 \pm 4) \implies F_1(2, 3), F_2(2, -5)$.
Eccentricity $e = 4/5 = 0.80$, Area $A = 15\pi \approx 47.12$.
Common Pitfalls & Major vs Minor Axis Inversion
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