Viviani's curve
Encyclopedia
In mathematics
, particularly geometry
, Viviani's curve, also known as Viviani's window, is a space curve
named after the Italian mathematician Vincenzo Viviani
, the intersection of a sphere
with a cylinder
that is tangent
to the sphere and passes through the center of the sphere.
given by
with the cylinder centered at
of radius 
given by
The resulting curve of intersection,
, can be parameterized by 
to give the parametric equation of Viviani's curve:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, Viviani's curve, also known as Viviani's window, is a space curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
named after the Italian mathematician Vincenzo Viviani
Vincenzo Viviani
Vincenzo Viviani was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo.-Biography:...
, the intersection of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
with a cylinder
Cylinder
Cylinder most commonly refers to:* Cylinder , a three-dimensional geometric shapeCylinder may also refer to:-Science and technology:* Cylinder , the space in which a piston travels in an engine...
that is tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to the sphere and passes through the center of the sphere.
Formula
The curve can be obtained by intersecting the unit sphere of radius given bywith the cylinder centered at
of radius given byThe resulting curve of intersection,
, can be parameterized by to give the parametric equation of Viviani's curve: