Ellipse Calculator Using 2 Points

Ellipse Equation:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Point 1 X-coordinate:

Point 1 Y-coordinate:

Point 2 X-coordinate:

Point 2 Y-coordinate:

Center X-coordinate (h):

Center Y-coordinate (k):

Ellipse Equation:

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1. What is an Ellipse Equation?

An ellipse is a closed curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. The standard equation of an ellipse centered at (h,k) is:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

2. How Does the Calculator Work?

The calculator uses two points on the ellipse and the center point to determine the semi-major (a) and semi-minor (b) axes:

\[ a = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \] \[ b = \sqrt{(x_2 - h)^2 + (y_2 - k)^2} \]

Where:

\( (x_1, y_1) \) — First point on ellipse
\( (x_2, y_2) \) — Second point on ellipse
\( (h, k) \) — Center of the ellipse
\( a \) — Semi-major axis length
\( b \) — Semi-minor axis length

3. Importance of Ellipse Calculation

Details: Ellipse equations are fundamental in geometry, physics (orbital mechanics), engineering, and computer graphics for modeling circular-like shapes with different radii in x and y directions.

4. Using the Calculator

Tips: Enter two distinct points that lie on the ellipse and the center coordinates. For best results, choose points that are not symmetric about the center.

5. Frequently Asked Questions (FAQ)

Q1: Why do I need more than 2 points for a complete ellipse?
A: Two points are insufficient to uniquely determine an ellipse - infinite ellipses can pass through two given points. This calculator assumes you know the center.

Q2: What's the difference between a circle and an ellipse?
A: A circle is a special case of an ellipse where a = b (both axes are equal length).

Q3: How can I find the foci of the ellipse?
A: The distance of each focus from the center is \( c = \sqrt{a^2 - b^2} \), located along the major axis.

Q4: What if my points don't lie on an ellipse?
A: The calculator will still compute values, but they won't represent a valid ellipse if the points don't satisfy the ellipse equation.

Q5: Can I use this for 3D ellipsoids?
A: No, this is for 2D ellipses only. Ellipsoids require a more complex 3D equation.