In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles (a diameter) of the sphere (see diagram). Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval. [1] [2]
The projection of Viviani's curve onto a plane perpendicular to the line through the crossing point and the sphere center is the lemniscate of Gerono. [3]
In 1692 Viviani tackled the task: Cut out of a half sphere (radius ) two windows, such that the remaining surface (of the half sphere) can be squared, i.e. a square with the same area can be constructed using only compasses and ruler. His solution has an area of (see below).
In order to keep the proof for squaring simple,
and
The cylinder has radius and is tangent to the sphere at point
Elimination of , , respectively yields:
The orthogonal projection of the intersection curve onto the
Representing the sphere by
and setting yields the curve
One easily checks that the spherical curve fulfills the equation of the cylinder. But the boundaries allow only the red part (see diagram) of Viviani's curve. The missing second half (green) has the property
With help of this parametric representation it is easy to prove the statement: The area of the half sphere (containing Viviani's curve) minus the area of the two windows is . The area of the upper right part of Viviani's window (see diagram) can be calculated by an integration:
Hence the total area of the spherical surface included by Viviani's curve is and the area of the half sphere () minus the area of Viviani's window is , the area of a square with the sphere's diameter as the length of an edge.
The quarter of Viviani's curve that lies in the all-positive quadrant of 3D space cannot be represented exactly by a regular bezier curve of any degree.
However, it can be represented exactly by a 3D rational bezier segment of degree 4, and there is an infinite family of rational bezier control points generating that segment.
One possible solution is given by the following five control points:
The corresponding rational parametrization is:
Subtracting 2× the cylinder equation from the sphere's equation and applying completing the square leads to the equation
which describes a right circular cone with its apex at , the double point of Viviani's curve. Hence
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.
In mathematics, an n-sphere or a hypersphere is a topological space that is homeomorphic to a standardn-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as
In physics, the Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.
In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.
A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.
In geometry, a nephroid is a specific plane curve whose name means 'kidney-shaped'.
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree . Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to and as
In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:
In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1
a and S1
b. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
a and S1
b each exists in its own independent embedding space R2
a and R2
b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics, which are well-defined at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
The goat problem is either of two related problems in recreational mathematics involving at least figuratively a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing the exterior of a circular area.
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.