A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can describe any conic section, simplifying both their representation and analysis.
The general polar equation of a conic is:
Here, the variable r is the distance from the origin to a point on the conic, �� is the polar angle, d is the distance from the focus to the directrix, and e is the eccentricity. The choice between cosine and sine depends on the orientation of the directrix, while the sign indicates its direction relative to the focus.
The specific nature of the conic section depends entirely on the value of the eccentricity. When the eccentricity is equal to one, the conic is called a parabola. When the eccentricity is greater than zero but less than one, the conic is an ellipse. When the eccentricity is greater than one, the conic is a hyperbola.
If the directrix is vertical, the equation involves the cosine function. A directrix to the right of the focus results in a plus sign with cosine, while a directrix to the left uses a minus sign. When the directrix is horizontal, the equation uses the sine function instead. A directrix above the focus leads to a plus sign with sine, and one below corresponds to a minus sign. These variations help specify the orientation and placement of the conic in the plane.
A conic section is the set of points for which the ratio of the distance to a fixed point, called the focus, and a fixed line, called the directrix, is constant.
This ratio is called eccentricity, and it determines the conic’s shape: an ellipse if between zero and one, a parabola if one, and a hyperbola if greater than one.
When the focus is at the origin in polar coordinates, a single polar equation describes all conics using the radial distance, polar angle, eccentricity, and distance to the directrix.
If the directrix is vertical, the equation uses cosine, which measures horizontal displacement in polar form.
A directrix to the right of the focus gives a plus sign, while one to the left gives a minus sign—indicating its position along the positive or negative horizontal axis.
If the directrix is horizontal, the equation uses sine, which measures vertical displacement in polar form.
A directrix above the focus gives a plus sign, while one below gives a minus sign—indicating if the directrix is on the positive or negative vertical axis.
Polar equations of conics also help design domes by defining curves that evenly distribute structural loads.
繁體中文
