1. What is the Foci and Vertices Calculator?

This calculator determines the positions of the foci and vertices of an ellipse given its semi-major and semi-minor axes. These parameters are fundamental in understanding the geometry of ellipses.

2. How Does the Calculator Work?

The calculator uses the ellipse formula:

Where:

• \( a \) — Semi-major axis length
• \( b \) — Semi-minor axis length
• \( c \) — Distance from center to each focus

Explanation: The vertices are located at the ends of the major axis (±a), while the foci are located along the major axis at a distance of ±c from the center.

3. Importance of Ellipse Parameters

Details: Knowing the foci and vertices is essential for understanding the shape and properties of an ellipse, which has applications in astronomy, physics, engineering, and many other fields.

4. Using the Calculator

Tips: Enter the semi-major (a) and semi-minor (b) axis lengths in any consistent units. The calculator will output the positions of vertices and foci in the same units.

5. Frequently Asked Questions (FAQ)

Q1: Which is the semi-major axis (a) and which is semi-minor (b)?
A: The calculator automatically uses the larger value as 'a' (semi-major) and the smaller as 'b' (semi-minor).

Q2: What if a equals b?
A: When a = b, the ellipse becomes a circle, and the foci coincide at the center (c = 0).

Q3: Can I use this for vertical ellipses?
A: Yes, the calculator works regardless of orientation. The major axis is always the longer one.

Q4: What are typical units for these measurements?
A: Any length units can be used (meters, centimeters, inches, etc.) as long as both axes are in the same units.

Q5: How accurate are the results?
A: Results are mathematically exact (limited only by floating-point precision) when accurate inputs are provided.