Conic curves include ellipses, hyperbolas and parabolas.
1.
Ellipse: A trajectory whose sum of the distances from a moving point to two fixed points is equal to a fixed length (the fixed length is longer than the distance between the two fixed points) is called an ellipse. Namely: {P|
|PF 1|+|PF2|=2a，
(2a & gt|F 1F2|)} .
2.
Hyperbola: The trajectory of a moving point with a fixed absolute value (the fixed value is less than the distance between two fixed points) is called hyperbola. That is, {p |||| pf1| | pf2 || = 2a,
(2a & lt|F 1F2|)} .
3.
Parabola: A trajectory with the same distance from a moving point to a fixed point and a fixed line is called parabola.
4.
Unified definition of conic section: The locus of a point whose ratio e between the distance to a fixed point and the distance to a fixed line is constant is called conic section. When 0
The origin of conic curve: circle, ellipse, hyperbola and parabola all belong to conic curve. As early as more than two thousand years ago, ancient Greek mathematicians were already familiar with them. Apollo, an ancient Greek mathematician, studied these curves with the method of plane truncated cone. Cut the cone with a plane perpendicular to the axis of the cone and you get a circle; Tilt the plane gradually to get an ellipse; When the plane is parallel to the generatrix of the cone, a parabola is obtained; When the plane is tilted a little more, you can get a hyperbola. Apollo once called ellipse "deficient curve", hyperbola "hypercurve" and parabola "homogeneous curve".
Parametric equation and rectangular coordinate equation of conic curve;
1) straight line
Parameter equation: x=X+tcosθ.
y=Y+tsinθ
(t is the parameter)
Cartesian coordinates: y=ax+b
2) Circle
Parametric equation: x=X+rcosθ
y=Y+rsinθ
(θ is a parameter.
)
Cartesian coordinates: x 2+y 2 = r 2
(r
Is the radius)
3) Ellipse
Parameter equation: x=X+acosθ.
y=Y+bsinθ
(θ is a parameter.
)
Cartesian coordinates (center as origin): x 2/a 2
+
y^2/b^2
=
1
4) hyperbola
Parametric equation: x=X+asecθ.
y=Y+btanθ
(θ is a parameter.
)
Cartesian coordinates (center as origin): x 2/a 2
-
y^2/b^2
=
1
(Opening direction is X axis)
y^2/a^2
-
x^2/b^2
=
1
(Opening direction is Y axis)
5) Parabola
Parametric equation: x = 2pt 2
y=2pt
(t is the parameter)
Cartesian coordinates: y = ax 2+bx+c
(the opening direction is the y axis,
a & lt& gt0
)
x=ay^2+by+c
(the opening direction is the x axis,
a & lt& gt0
)
The unified polar coordinate equation of conic curve (quadratic noncircular curve) is
ρ=ep/( 1-e cosθ)
Where e stands for eccentricity and p is the distance from the focus to the directrix.