Question

Give a note on the properties of ellipse in computer graphics

Answer 1

It is used to make circles and logos

Answer 2

An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

An ellipse is a set of points, such that for any point {\displaystyle P} of the set, the sum of the distances {\displaystyle |PF_{1}|,\ |PF_{2}|} to two fixed points {\displaystyle F_{1}}, {\displaystyle F_{2}}, the foci, is constant, usually denoted by {\displaystyle 2a,\ a>0\ .} In order to omit the special case of a line segment, one assumes {\displaystyle 2a>|F_{1}F_{2}|.} More formally, for a given {\displaystyle a}, an ellipse is the set {\displaystyle E=\{P\in \mathbb {R} ^{2}\mid |PF_{2}|+|PF_{1}|=2a\}\ .}

The midpoint {\displaystyle C} of the line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis, and the line perpendicular to it through the center is called the minor axis. It contains the vertices {\displaystyle V_{1},V_{2}}, which have distance {\displaystyle a} to the center. The distance {\displaystyle c} of the foci to the center is called the focal distance or linear eccentricity. The quotient {\displaystyle {\tfrac {c}{a}}} is the eccentricity {\displaystyle e}.

The case {\displaystyle F_{1}=F_{2}} yields a circle and is included.

The equation {\displaystyle |PF_{2}|+|PF_{1}|=2a} can be viewed in a different way (see picture):
If {\displaystyle c_{2}} is the circle with midpoint {\displaystyle F_{2}} and radius {\displaystyle 2a}, then the distance of a point {\displaystyle P} to the circle {\displaystyle c_{2}} equals the distance to the focus {\displaystyle F_{1}}:

{\displaystyle |PF_{1}|=|Pc_{2}|.}

{\displaystyle c_{2}} is called the circular directrix (related to focus {\displaystyle F_{2}}) of the ellipse.[1][2] This property should not be confused with the definition of an ellipse with help of a directrix (line) below.

Using Dandelin spheres one proves easily the important statement:

Any plane section of a cone with a plane, which does not contain the apex and whose slope is less than the slope of the lines on the cone, is an ellipse.