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LIFT OF A PLANE CURVE ON A SURFACE

The curve
of the plane xOy can be lifted on the surface
as the curve .
The curve can be lifted on the surface of revolution
as the curve . |

The lift of a plane curve on a surface is the intersection between the cylinder built on the curve perpendicularly to its plane and the surface. Therefore, it is the maximal curve traced on the surface for which the orthogonal projection is the plane curve.

Examples:

- the lifts of straight lines on a
surface are the plane curves traced on the surface.

- the lifts of circles are the cylindrical
curves.

- the clelias
are lifts of roses
on a sphere, whereas the basins
are lifts of them on hyperbolic paraboloids, and conical
roses, lifts on cones of revolution.

- The conical
helix is a lift of a logarithmic spiral, whereas the conical
spiral of Pappus is a lift of the Archimedean spiral.

- the spherical
helices are lifts of epicycloids.

- the helices
of the hyperbolic paraboloid (placed vertically) are lifts of involutes
of circles.

- the geodesics
of the paraboloid of revolution are lifts of hypercycloids.

Conical lifts of Archimedean spirals on the left, logarithmic spirals on the right. | ||

Lifts on a hyperbolic paraboloid of Archimedean spirals on the left, logarithmic spirals on the right. |

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© Robert FERRÉOL 2018