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The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1) 2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1.
The blue area above the x-axis may be specified as positive area, while the yellow area below the x-axis is the negative area. The integral of a real function can be imagined as the signed area between the x {\displaystyle x} -axis and the curve y = f ( x ) {\displaystyle y=f(x)} over an interval [ a , b ].
Let φ 1 = 0, φ 2 = 2π; then the area of the black region (see diagram) is A 0 = a 2 π 2, which is half of the area of the circle K 0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a 2 π 2. Hence: The area between two arcs of the spiral after a full turn equals the area of the circle ...
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and
Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and then we have Case 7. There are either one or two solutions. Case 4: two angles and an included side given (ASA). The four-part cotangent formulae for sets (cBaC) and (BaCb) give c and b, then A follows from the sine rule. Case 5: two angles and an opposite side given ...
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
When rectified, the curve gives a straight line segment with the same length as the curve's arc length. Arc length s of a logarithmic spiral as a function of its parameter θ. Arc length is the distance between two points along a section of a curve.