07. Ellipse and other Curves

Dated: 15-04-2025

Ellipse

\[|\overline{O V_2}| = |\overline{O V_1}| = a\]

\[|\overline{O V_2^\prime}| = |\overline{O V_1^\prime}| = b\]

\[|\overline{O F_2}| = |\overline{O F_1}| = c\]

\[|r_1 + r_2| = 2c\]

History

The ellipse was first studied by Menaechmus.
It was further investigated by Euclid and later named by Apollonius.
The concepts of focus and conic section directrix of an ellipse were studied by Pappus.
In 1602, Kepler believed that the orbit of Mars was oval.
He later discovered it was an ellipse with the Sun at one focus.
Kepler introduced the word "focus" and published his findings in 1609.
In 1705, Halley showed that the comet now named after him follows an elliptical orbit around the Sun.
An ellipse rotated about its minor axis gives an oblate spheroid.
An ellipse rotated about its major axis gives a prolate spheroid.

\[\frac {x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Assuming \(a > b\), \(b\) is called the semi minor axis and \(a\) is called the semi major axis.
In general,
Equation of an ellipse centered at \((x_0, y_0)\) is

\[\frac {(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} = 1\]

Ellipse Drawing Techniques

\[\frac {x^2}{a^2} + \frac{y^2}{b^2} = 1\]

\[\implies b^2x^2 + a^2 y^2 = a^2b^2\]

\[f_{\text{ellipse}} = b^2x^2 + a^2 y^2 - a^2b^2\]

Let's assume \(a > b\) and accordingly, \(r_x = a\) and \(r_y = b\).

\[f_{\text{ellipse}} = r_y^2x^2 + r_x^2 y^2 - r_x^2r_y^2\]

\[f_{\text{ellipse}} = \begin{cases} < 0 & \text{if } (x, y) \text{ is inside the ellipse boundary} \\ = 0 & \text{if } (x, y) \text{ is on the ellipse boundary} \\ > 0 & \text{if } (x, y) \text{ is outside the ellipse boundary} \\ \end{cases} \]

Starting from \((0, r_y)\), we step using \(x\) where \(y\) is determined by decision function.
When slope¹ \(m = -1\), we switch the stepping from \(x\) to \(y\).
The slope¹ of the ellipse can be figured out by using

\[\frac{dy}{dx} = - \frac{r_y^2x^2}{r_x^2y^2}\]

The boundary where we switch stepping, is determined by \(\frac {dy}{dx} = -1\).
This gives us

\[r_y^2x^2 = r_x^2y^2\]

we will switch the stepping whenever

\[r_y^2x^2 \ge r_x^2y^2\]

Similar to previous lecture, for region 1, if \(P_k < 0\) then \(y_{k + 1} = yk\) otherwise, \(y_{k + 1} = y_k - 1\).

\[P_{k + 1} = P_k + 2r_y^2(x_k + 1) + r_y^2 \quad \text{where } P_k < 0\]

\[P_{k + 1} = P_k + 2r_y^2(x_k + 1) + r_y^2 - 2r_x^2(y_k - 1)\quad \text{where } P_k > 0\]

For our stating point \(P_0\), we will put \((0, r_y)\),

\[P_0 = r_y^2 - r_x^2r_y + \frac 1 4 r_x^2\]

Similarly, for region 2

\[P_{k + 1}^\prime = P_k^\prime - 2 r_x^2(y_k + 1) + r_x^2 \quad \text{where } P_k^\prime > 0\]

\[P_{k + 1}^\prime = P_k^\prime + 2 r_y^2(x_k + 1) + r_x^2 - 2r_x^2y_k \quad \text{where } P_k^\prime < 0\]

Our starting point \(P_0^\prime\) will be the last point computed in region 1.

\[P_0^\prime = r_y^2 \left(x_0 + \frac 1 2\right) + r_x^2(y_0 - 1)^2 - r_x^2r_y^2\]

MidPointEllipse(xcenter, ycenter, rx, ry):
    x = 0
    y = ry

    do {
        DrawSymmetricPoints (xcenter, ycenter, x, y)

        R1_P0 = ry**2 - rx**2 * ry + rx**2 * 1 / 4

        if (R1_Pk < 0) {
            R1_Pk_next = R1_Pk + 2ry**2 * (xk + 1) + ry**2
        }
        else {
            R1_Pk_next = R1_Pk + 2ry**2 * (xk + 1) + ry**2 - 2rx**2 * (yk - 1)
        }

        R2_P0 = ry**2 * (x0 + 1 / 2) + rx**2 * (y0 - 1)**2 - rx**2 * ry**2

        y++

        if (R2_Pk > 0) {
            R2_Pk_next = R2_PK - 2rx**2 * (yk + 1) + rx**2
        }
        else {
            R2_Pk_next = R2_PK - 2rx**2 * yk + rx**2 + 2ry**2 (xk + 1)
        }
        x++
    } while (ry**2 * x**2 >= rx**2 * y**2);
}

Other Curves

Various curves are useful for

Object Modeling
Animation path specifications
Data and function² graphing
Graphics applications

Commonly encountered curves include

Conics
Trigonometric
Exponential
Probability distributions
General polynomials
Splines

Conic Sections

The general equation for a conic section is

\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]

Type of the section can be found using

\[B^2 - 4AC\]

\(< 0\)
- ellipse
- circle³
- point
- no curve
\(= 0\)
- parabola
- 2 parallel lines
- 1 line
- no curve
\(> 0\)
- hyperbola
- 2 intersecting lines

	Circle	Ellipse	Parabola	Hyperbola
Equation (horizontal vertex)	\(x^2 + y^2 = r^2\)	\(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1\)	\(4px = y^2\)	\(\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\)
Equations of Asymptotes				\(y = \pm x \left(\frac b a\right)\)
Equation (vertical vertex)	\(x^2 + y^2 = r^2\)	\(\frac{y^2}{a^2}+\frac{x^2}{b^2} = 1\)	\(4py = x^2\)	\(\frac{y^2}{a^2}-\frac{x^2}{b^2} = 1\)
Equations of Asymptotes				\(x = \pm y \left(\frac b a\right)\)
Variables	\(r =\) circle radius	\(a=\) major radius, \(b=\) minor radius, \(c=\) distance from `center` to `focus`	\(p =\) distance from `vertex` to `focus` or `directrix`	\(a=\frac 1 2\) length of major axis, \(b=\frac 1 2\) length of minor axis, \(c=\) distance from `center` to `focus`
Eccentricity	\(0\)		\(\frac c a\)	\(\frac c a\)
Relation to `Focus`	\(p = 0\)	\(a^2 - b^2 = c^2\)	\(p = p\)	\(a^2 + b^2 = c^2\)
Definition: locus of `points` meeting the condition	distance to the `origin` is `constant`	sum of distances to each `focus` is `constant`	distance to `focus` = distance to `directrix`	difference between `distances` to each `foci` is constant

Hyperbola

Set⁴ of all points, whose difference of distances from 2 fixed points (the foci) is constant, is called a hyperbola.

\[\overline{V_1V_2} = \text{Transverse Axis}\]

Equation

Horizontal Transverse Axis

Centered at \((0, 0)\)

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

Centered at \((h, k)\)

\[\frac{(x - h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]

Vertical Transverse Axis

Centered at \((0, 0)\)

\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]

Centered at \((h, k)\)

\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]

If \(c = |\overline{FO}|\) and \(a = |\overline{VO}|\) then

\[b^2 = c^2 - a^2\]

Eccentricity \(e\) is given by

\[e = \frac c a\]

For hyperbola, \(e > 1\) and for ellipse, \(1 \ge e > 0\)

The coordinates of the corners of the rectangle shown in the figure are

\[(h \pm a, k \pm b)\]

and the dotted lines passing through these corners are called asymptotes, bounding both branches of the hyperbola.

Slope¹ is given by

\[\frac b a\]

The equations for the asymptotes are

For horizontal hyperbola

\[y = k \pm \frac b a (x - h)\]

For vertical hyperbola

\[y = k \pm \frac a b (x - h)\]

Also, the line⁵ joining both ends of the rectangle, passing through the center and perpendicular to the axis is called conjugate axis.

Parabola

It is defined as set⁴ of points which are equidistant from a line⁵ called directrix and a fixed point called focus which isn't on directrix.
The midpoint between directrix and focus is called the vertex.
The line passing through focus and vertex is called the axis.

Although parabola looks like one of the branches of hyperbola, it is not quite the case.

Equations for parabolas with vertex at \((h, k)\) are given by

For horizontal parabola

\[(y - k)^2 = 4p(x - h)\]

For vertical parabola

\[(x - h)^2 = 4p(y - k)\]

Rotations of Axes

The \(B\) term in the general equation

\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]

controls rotation.

Animated Applications

Ellipses, hyperbolas and parabolas are useful for describing orbital motions in context of following forces.

Gravitational
Electromagnetic
Nuclear

Parabolic Reflectors

Car headlights
Telescope mirrors
Antennas

Elliptical Orbits

Sun being one of the foci was noted by Apollonius in 2nd century and was later studied by Kepler rigorously.
The eccentricity of the following orbits are

\(0.0167\) for earth
\(0.2481\) for pluto (highest in all planets)

Whispering Galleries

In United States capital, St. Paul's Cathedral has one of the rooms with an elliptical ceiling which allows two people standing at different foci of the ellipse hear each other very closely.

Polynomials and Spline Curves

\[y = \sum_{i = 0}^n a_ix^i\]

\[y = a_0x^0 + a_1x^1 + \ldots + a_nx^n\]

Applications involve

Design of object shapes
Specifications of animation paths
Graphing of data trends in a discrete set of data points

To design object shapes or motion paths, a few key points are chosen to outline the curve, then polynomial fitting is applied.
A common approach uses cubic polynomial segments between each pair of points, defined parametrically as:

\[x(u)=ax0+ax1u+ax2u2+ax3u3x(u) = a_{x0} + a_{x1}u + a_{x2}u^2 + a_{x3}u^3\]

\[y(u)=ay0+ay1u+ay2u2+ay3u3y(u) = a_{y0} + a_{y1}u + a_{y2}u^2 + a_{y3}u^3\]

where \(u \in [0,1]u \in [0, 1]\).

References

Read more about slopes. ↩↩↩
Read more about functions. ↩
Read more about circle. ↩
Read more about sets. ↩↩
Read more about lines. ↩↩