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How does the angle of a cutting plane determine whether a conic is a circle, ellipse, parabola, or hyperbola?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, a conic section is defined by the intersection of a plane and a right circular cone. The specific type of curve produced depends on the angle of the cutting plane relative to the cone's axis or its slant height.1

The classification is determined by the relationship between the cutting plane and the cone:

Circle: Formed when the cutting plane is perpendicular to the axis of the cone.1
Ellipse: Formed when the plane cuts through the cone at an angle such that it intersects only one nappe of the cone, creating a closed curve.1
Parabola: Formed when the cutting plane is parallel to the slant height (the side) of the cone, resulting in a curve that is open at one end.1
Hyperbola: Formed when the plane cuts through both nappes of the cone, creating two separate, open branches.21

From a projective geometry perspective, these curves can be viewed as images of a circle projected onto a plane. The orientation of the plane relative to the cone's apex and its base determines whether the resulting image is a circle, ellipse, parabola, or hyperbola.3