Conic Sections and Their 3D Shapes — Circles, Ellipses, Parabolas, and Hyperbolas in Two and Three Dimensions

hari@dasarpai.com (Dr. Hari Thapliyaal) — Mon, 24 Aug 2026 00:00:00 +0000

Conic Sections and Their 3D Shapes
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Circles, ellipses, parabolas, and hyperbolas in two and three dimensions
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A circle is not an isolated shape. It is the simplest member of a family: rotate it around an axis and you get a sphere; slide the same cross-section along a straight line and you get a cylinder. The same pattern repeats for ellipses, parabolas, and hyperbolas — each 2D curve has a natural 3D surface of revolution and a natural cylindrical extrusion.

This article is a shape-and-formula guide. For each family you get:

the standard equation in the plane,
the 3D surface you obtain by rotation,
the 3D surface you obtain by extrusion along the \(z\)-axis,
everyday names so you can connect the formula to objects you already know,
a labeled illustration to fix the picture in memory.

We work in Cartesian coordinates \((x,y)\) in the plane and \((x,y,z)\) in space. Parameters \(a,b,c,p,r>0\) unless noted otherwise.

Geometry on dasarpAI

Conic Sections and Their 3D Shapes — Circles, Ellipses, Parabolas, and Hyperbolas in Two and Three Dimensions

Conic Sections and Their 3D Shapes #

Circles, ellipses, parabolas, and hyperbolas in two and three dimensions #

Conic Sections and Their 3D Shapes
#

Circles, ellipses, parabolas, and hyperbolas in two and three dimensions
#