arc length

In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short e ...

Including:

• Ellipse - Parametrisation
• Ellipse - Eccentricity
• Ellipse - Semi-latus rectum and polar coordinates
• Ellipse - Area
• Ellipse - Circumference
• Ellipse - Stretching and Projection
• Ellipse - Reflection property
• Ellipse - Ellipses in physics
• Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia - Ellipse

In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: (x2 + y2)2 = a2(x2 − y2) Graphing this equation produces a curve similar to . The curve has become a symbol of infinity and is widely used in math. The symbol itself is sometimes referred to as the lemniscate. Its Unicode represe ...

Including:

• Lemniscate - Other equations
• Lemniscate - Arc length and elliptic functions

Read more here: » Lemniscate: Encyclopedia - Lemniscate

Gauss's constant may be used as a closed-form expression for the Gamma function at argument 1/4: and since π and Γ(1/4) are algebraically independent, Gauss's constant is transcendental. Gauss's constant - Lemniscate constants. Gauss's constant may be used in the definition of the lemniscate constants, the first of which is: and the second constant: which arise in finding the arc length of a lemniscate. Gauss's constant - Calculation. A formula for G in terms of Jacobi theta fu ...

See also:

Gauss's constant, Gauss's constant - Uses, Gauss's constant - Lemniscate constants, Gauss's constant - Calculation, Gauss's constant - External links

Read more here: » Gauss's constant: Encyclopedia II - Gauss's constant - Uses

A semicubical parabola is a curve defined parametrically as The parameter can be removed to yield the equation ...

See also:

Semicubical parabola, Semicubical parabola - Definition, Semicubical parabola - Properties, Semicubical parabola - History

Read more here: » Semicubical parabola: Encyclopedia II - Semicubical parabola - Definition

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid. For example, the surface area of the torus with minor radius r and major radius R is ...

See also:

Pappus's centroid theorem, Pappus's centroid theorem - The first theorem, Pappus's centroid theorem - The second theorem

Read more here: » Pappus's centroid theorem: Encyclopedia II - Pappus's centroid theorem - The first theorem

Elliptic integrals are often expressed as functions of a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but can be confusing due to their different appearance. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments: k the elliptic modulus m=k2 the parameter α the modular angle, See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Notation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametr ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Parametrisation

A lemniscate may also be described by the polar equation r2 = a2cos2φ or the bipolar equation ...

See also:

Lemniscate, Lemniscate - Other equations, Lemniscate - Arc length and elliptic functions

Read more here: » Lemniscate: Encyclopedia II - Lemniscate - Other equations

The complete elliptic integral of the second kind E is defined as Or if 0 ≤ k ≤ 1: ...

See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Complete elliptic integral of the second kind

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement or where c (the linear eccentricity of the ellipse) equals the distance from the cen ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Eccentricity

A special case of the semicubical parabola can be used to define the evolute of the parabola: Expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola: ...

See also:

Semicubical parabola, Semicubical parabola - Definition, Semicubical parabola - Properties, Semicubical parabola - History

Read more here: » Semicubical parabola: Encyclopedia II - Semicubical parabola - Properties

The complete elliptic integral of the first kind K is defined as and can be computed in terms of the arithmetic-geometric mean. It can also be calculated as Or in form of integral of sine, when 0 ≤ k ≤ 1 The complete elliptic integral of the first kind is sometimes called the quarter period. ...

See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Complete elliptic integral of the first kind

The incomplete elliptic integral of the third kind Π is or or The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value Π(1;π / 2 | m) is infinite, for any m. ...

See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Incomplete elliptic integral of the third kind

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Ellipses in computer graphics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [1]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is nega ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Ellipses in physics

The semi-latus rectum of an ellipse, usually denoted (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity, . In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane giv ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Semi-latus rectum and polar coordinates

The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as Equivalently, using alternate notation, where it is understood that when there is a vertical bar used, the argument following the vertical bar is the parameter (as defined above), and, when a backslash is used, it is followed by the modular angle. Note that F(x;k) = u with u as defined ...

See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the second kind E is Equivalently, using alternate notation, Additional relations include ...

See also:

Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History

Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Incomplete elliptic integral of the second kind

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: A good approximation is Ramanujan's: which can also be written as: More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc le ...

See also:

Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

Read more here: » Ellipse: Encyclopedia II - Ellipse - Circumference