What Is an Ellipse?

Ellipse is the locus of moving point P whose sum of the distances to the fixed points F1 and F2 in the plane is equal to a constant (greater than | F1F2 |). The mathematical expression is: | PF1 | + | PF2 | = 2a (2a> | F1F2 |). ^[1]

In mathematics, an ellipse is a curve in a plane surrounding two focal points, so that for each point on the curve, the sum of the distances to the two focal points is constant. It is therefore a generalization of a circle, which is a special type of ellipse with two focal points at the same location. The shape of an ellipse (how "stretched") is represented by its eccentricity. For an ellipse, it can be any number from 0 (the limit case of a circle) to any number that is close to but less than 1.

An ellipse is a closed conical section: a plane curve intersected by a cone and a plane. Ellipses have many similarities to the other two forms of conical sections: parabola and hyperbola, both of which are open and unbounded. The cross section of a cylinder is elliptical unless the section is parallel to the axis of the cylinder.

An ellipse can also be defined as a set of points such that the ratio of the distance of each point on the curve to the distance of a given point (called the focal point) to the distance of the same point on the curve is given by a given row (called directrix) . The ratio is called elliptical

An elliptical mirror (a three-dimensional figure formed by rotating the ellipse 180 degrees with the ellipse's long axis as its axis, and all the inner surfaces are made of reflective surfaces, hollow) can reflect all the light emitted from one focal point to another focal point; elliptical

Example: There is one

Ellipse Freehand Method I

(1): Draw the long axis AB, the short axis CD, and AB and CD are perpendicular to each other and bisected at O point.

(2): Connect AC.

(3): Use O as the center of the circle and OA as the radius to make the OC extension line at point E.

(4): Use C as the center and CE as the radius to make an arc and AC at point F.

(5): Make the vertical bisector of AF cross the CD extension line at point G and cross AB at point H.

(6): intercept the symmetry points H ', G' of O to point O ', G' : H, H 'are the long axis centers, and HA and H'B are the radii respectively; G, G' is the center of the short axis Make circles with GC and G'D as the radii, respectively.

Use a wire or a thin copper wire, pencil, two pins or pins to draw an ellipse: first draw the long and short axis cross lines, and use the dot as the center on the long axis to find two points larger than the short axis radius. One point is fixed with a pin or a pin first, and the line at the other point is not fixed. Use the pen to hold the line to find the 4 vertices of the long and short axes. This step requires multiple positioning until all of them coincide with the vertex. After fixing these 2 points, use a pen to hold the line and draw the ellipse directly :) It is best to use thin copper wire, because the elasticity of the line is not necessarily accurate.

Ellipse Freehand Method II

The focal length of the ellipse FF' (Z) is defined as the long axis X (ab) and the short axis Y (cd) of the known ellipse. Use the long axis A as the center of the circle and the short axis Y as the radius to draw an arc. The line segment tangent to the arc drawn from the other point B of the axis is the ellipse focal length. The verification formula is 2 {(Z / 2) ^ 2 + (Y / 2) ^ 2} + Z = X + Z (in-plane and The sum of the distances between the two fixed points F and F 'is equal to the constant 2a (2a> | FF' |). The locus of the moving point P is called an ellipse), which can be transformed into z = x ^ 2-y ^ 2 (x> y> 0 ). Points F and F 'at both ends of Z are fixed points. Take the line with the smaller ductile shear expansion coefficient, the better, the surrounding line segment AF 'or FB line segment any group as the length, with the length as the fixed triangle perimeter, and F, F' as the fixed points, take the first part of the triangle Drawing an arc at three points for the moving point constitutes the ellipse. ^[4]

Oval diagram

Ellipse Freehand Method III

Loop length

. According to the graphic characteristics of the ellipse, a circular line is used to represent the distance relationship between the moving point and the focal point to form a unified circular circular line drawing method. Specific method introduction:

(1) The drawing tools are pen, pin, ruler, and loop line. (Ring production: take a length (30-50cm) and a small thickness of moderately flexible elastic wire, a section of 8mm long thin empty plastic pipe, the flexible wire runs through the plastic pipe, the plastic pipe clamps the flexible wire, but Force can be twitched to form a circular line that can shrink and grow).

(2) Make various fixed and moving points on the drawing plane.

(3) Put the pins upright and fix them at fixed points respectively;

(4) Put a circular wire of a suitable length on the outside of the pin, and straighten the loop line from the inside to the outside, and adjust the length of the loop line so that the pen tip just falls on the moving point;

(5) Move the paintbrush once to make various round shapes.

The biggest feature of the circle line mapping method is that the distance relationship between the circular moving point and the focus is connected in a circle line without being affected by the number of focus points. The circle line can accommodate any number of focus points. Multifocal circles with more than 3 focal points provide an effective method. The circle drawing method is a continuous moving drawing method, and is suitable for drawing circles, ellipses and ovals of different sizes.

If this method is used to draw an ellipse with a semi-major axis a and a semi-minor axis b, then

, Loop length