Conic Sections Calculator

Conic sections are curves formed by the intersection of a plane and a double-napped cone. The specific curve is determined by its eccentricity (e), which measures how much the conic section deviates from being circular. For a circle, e = 0; for an ellipse, 0 < e < 1; for a parabola, e = 1; and for a hyperbola, e > 1. Key mathematical formulas govern these shapes. For instance, the eccentricity of a horizontal ellipse is calculated using the formula e = sqrt(1 - (b² / a²)), where a represents the semi-major axis and b represents the semi-minor axis. Similarly, linear eccentricity (c) relates the focal distance from the center, while the semi-latus rectum defines the width of the section passing through a focus perpendicular to the major axis.

Let us analyze a horizontal ellipse with a major axis length of 8 units and a minor axis length of 4 units. First, we identify our semi-axes: the semi-major axis a = 4 and the semi-minor axis b = 2. To find the eccentricity, we apply the formula e = sqrt(1 - (b² / a²)). Substituting our values gives e = sqrt(1 - (2² / 4²)) = sqrt(1 - (4 / 16)) = sqrt(1 - 0.25) = sqrt(0.75) ≈ 0.866. Next, we calculate the linear eccentricity using c = sqrt(a² - b²), which yields sqrt(16 - 4) = sqrt(12) ≈ 3.464. Finally, the semi-latus rectum is found via b² / a = 4 / 4 = 1. These precise metrics completely define the shape and focal properties of the ellipse.

When inputting values into your conic calculations, always double-check whether your shape is oriented horizontally or vertically, as swapping major and minor axis inputs will invert your eccentricity results. Ensure consistency in your units of measurement across all variables. Additionally, remember that for hyperbolas, the relationship between the axes differs from ellipses; the linear eccentricity formula incorporates an addition sign instead of subtraction under the radical.