Conic curve is one of the main objects of differential geometry research, which can be regarded as the trajectory of spatial particle motion. In solid geometry, a cone is a geometric body formed by rotating the other two sides 360 degrees with the straight line on the right side of a right triangle as the rotation axis.
The rotation axis is called the axis of a cone, the surface that rotates perpendicular to the axis is called the bottom surface of the cone, and the surface that rotates not perpendicular to the axis is called the side surface of the cone. No matter where you rotate, the side that is not perpendicular to the axis is called the generatrix of the cone.
Conic curves include ellipse (circle is a special case of ellipse), parabola and hyperbola. The ancient Greek mathematicians who originated more than 2000 years ago first began to study conic curves.
The (incomplete) unified definition of conic: the locus of a point whose ratio of the distance r from a point to a point on a plane to the distance d from a point to a straight line is constant e=r/d is called conic. When e> 1 is hyperbola, when e= 1 is parabola, and when 0 is less than e, it is ellipse. The fixed point is called the focus of the conic, the fixed line is called the directrix, and e is called the eccentricity.
origin
More than 2000 years ago, ancient Greek mathematicians first began to study conic curves, and achieved a lot of results. Apollonius, 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 will get a circle.
Tilt the plane gradually to get an ellipse; When the plane is inclined to "and only parallel to a generatrix of the cone", a parabola is obtained; When the plane parallel to the axis of the cone is cut, a hyperbola can be obtained (when the conical surface is replaced by the corresponding conical surface, a hyperbola can be obtained).
Apollo once called ellipse "deficient curve", hyperbola "hypercurve" and parabola "homogeneous curve". In fact, in his works, Apolloni has obtained all the properties and results of conic curves in today's high school mathematics by means of pure geometry.
Crane focus chord
Similar to a circle, the line connecting any two points on a conic curve is called a chord; The string passing through the focal point is called focus chord. Focus chord parallel to the directrix is called path, and it is also called positive focus chord in physics.
Conic curves are smooth, so there are concepts of tangent and normal. For the same ellipse or hyperbola, the combination of two focus directrix can get it. Therefore, an ellipse and a hyperbola have two focuses and two directrix. A parabola has only one focus and one directrix.
A conic curve is an axisymmetric figure, and the axis of symmetry is a straight line passing through the focus and perpendicular to the directrix. In the case of ellipse and hyperbola, a straight line passes through two focal points, which is called the focal axis of conic curve. For ellipses and hyperbolas, there is also symmetry about the midline of the focal line, so ellipses and hyperbolas have two symmetry axes.