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A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances from two fixed points is the square of half the distance between the points. It is a particular case of the Cassinian oval and resembles a figure 8. When the line joining the two fixed points is the axis of x and the middle point of this line is the origin, the Cartesian equation is the fourth degree equation, (((x^2)+(y^2))^2)=2(a^2)((x^2)-(y^2)). The polar equation is (ℽ^2) = 2(a^2)cos(2θ). The locus of the feet of the perpendiculars from the center of an equilateral hyperbola to its tangents is a lemniscate. The name lemniscate is sometimes given to any crunodal symmetric quartic curve having no infinite branch. The name is also sometimes given to a general class of curves derived from other curves in the way that the above is derived from the equilateral hyperbola. With these more general definitions of the lemniscate the above curve is called the lemniscate of Bernoulli.

Lemniscate

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances…

An Amsler-type polar planimeter that is being used to measure the area of a load graph.

Polar Planimeter

An Amsler-type polar planimeter that is being used to measure the area of a load graph.

The polar bear, Plantigrada, is part of the subdivision Carnivora, which includes other carnivorous animals, like the bear and raccoon, that walk with the heel up on the ground. a, femur, or thigh; b, tibia, or leg; c, tarsus and metatarsus, or foot; d, calx, or heel; e, planta, or sole; f, digits, or toes.

Polar Bear Leg

The polar bear, Plantigrada, is part of the subdivision Carnivora, which includes other carnivorous…

The anatomy of a polar bear's leg. a, femur (thigh); b, tibia (leg); c, tarsus and metatarsus (foot); d, calx (heel); e, planta (sole); f, digits (toes).

Polar Bear Leg

The anatomy of a polar bear's leg. a, femur (thigh); b, tibia (leg); c, tarsus and metatarsus (foot);…

Showing how to find points on a coordinate plane. Also, shows an example of the midpoint formula.

Point Ratio

Showing how to find points on a coordinate plane. Also, shows an example of the midpoint formula.

Polar coordinate example. Ex. (4, 60 degrees)

Coordinate Example

Polar coordinate example. Ex. (4, 60 degrees)

Changing from polar to rectangular coordinates.

Rectangular Coordinates

Changing from polar to rectangular coordinates.

Finding the distance between two points using polar coordinates.

Distance Between

Finding the distance between two points using polar coordinates.

Finding the area of a triangle in terms of polar coordinates of its three vertices.

Triangle Area

Finding the area of a triangle in terms of polar coordinates of its three vertices.

If the chords of a circle are drawn through a fixed point, then the points of intersection of the pairs of tangents at the extremities of the chords will all lie on a fixed straight line.

Polar Property

If the chords of a circle are drawn through a fixed point, then the points of intersection of the pairs…

Polar property example.

Example

Polar property example.

Constructing the polar of a given point P with respect to a circle.

Polar of Point

Constructing the polar of a given point P with respect to a circle.

Finding the polar equation of a straight line.

Polar Equation

Finding the polar equation of a straight line.

Finding the polar equation of a straight line by passing through two given points.

Straight Line

Finding the polar equation of a straight line by passing through two given points.

Finding the equation of a line with polar coordinates.

Polar Coordinates

Finding the equation of a line with polar coordinates.