Linear Eccentricity of Ellipse given Eccentricity and Semi Minor Axis Calculator

Linear Eccentricity of Ellipse Formula:

\[ c = \frac{e \cdot b}{\sqrt{1 - e^2}} \]

Linear Eccentricity of Ellipse (c):

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1. What is Linear Eccentricity of Ellipse?

Linear Eccentricity of an Ellipse is the distance from the center of the ellipse to either of its two foci. It's an important geometric property that helps define the shape and characteristics of an ellipse.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ c = \frac{e \cdot b}{\sqrt{1 - e^2}} \]

Where:

\( c \) — Linear Eccentricity of Ellipse (meters)
\( e \) — Eccentricity of Ellipse (unitless, 0 ≤ e < 1)
\( b \) — Semi Minor Axis of Ellipse (meters)

Explanation: This formula calculates the distance from the center to the foci using the eccentricity and semi-minor axis of the ellipse.

3. Importance of Linear Eccentricity Calculation

Details: Linear eccentricity is crucial in understanding the geometric properties of ellipses, which have applications in astronomy (planetary orbits), engineering (arch design), and physics (orbital mechanics).

4. Using the Calculator

Tips: Enter eccentricity (between 0 and 1) and semi-minor axis (positive value). The calculator will compute the linear eccentricity of the ellipse.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of possible values for eccentricity?
A: Eccentricity ranges from 0 (circle) to values approaching 1 (highly elongated ellipse).

Q2: How does linear eccentricity relate to the semi-major axis?
A: Linear eccentricity is also equal to \( c = \sqrt{a^2 - b^2} \), where a is the semi-major axis.

Q3: What happens when eccentricity equals 0?
A: When e=0, the ellipse becomes a circle, and the linear eccentricity becomes 0 (foci coincide with the center).

Q4: Can eccentricity be exactly 1?
A: No, eccentricity of 1 represents a parabola, not an ellipse. The formula is undefined at e=1.

Q5: What are practical applications of this calculation?
A: This calculation is used in astronomy for orbital calculations, in engineering for arch and bridge design, and in optics for elliptical mirror designs.