3.5 Contents

1. Universal Properties

2. Circles

3. Ellipses

4. Parabolas

5. Hyperbolas

6. Rotations 🔧

3.5.1 Universal Properties

Two-dimensional subsets of a 3D (double) cone surface in which the shapes are determined by the intersection of a 2D plane

General Equation

$$A⋅x^2+B⋅x⋅y+C⋅y^2+D⋅x+E⋅y+F=0$$ All conic sections will be represented without rotation until the section on rotations. In other words, $B=0$.

Discriminant & Related Properties

$$\Delta=B^2-4⋅A⋅C$$ Assuming no other values:

• If $B=0$, the function is even either vertically or horizontally (not rotated)
• If $Δ \lt 0$, the function is an ellipse
• If $Δ \lt 0$, $B=0$, and $A \lt C$, the function is an ellipse with a horizontal major axis
• If $Δ \lt 0$, $B=0$, and $A \gt C$, the function is an ellipse with a vertical major axis
• If $Δ \lt 0$, $B=0$, and $A=C$, the function is a circle
• If $Δ \lt 0$, $A=C$, $B=D=E=0$, the function is a circle centered at the origin
• If $Δ=0$, the function is a parabola
• If $C=D=F=0$, the function is a vertical parabola with the vertex at the origin
• If $A=E=F=0$, the function is a horizontal parabola with the vertex at the origin
• If $D=E=0$, the function is either an ellipse or hyperbola centered at the origin
• If $Δ \gt 0$, the function is a hyperbola
• If $Δ \gt 0$ and $A+C=0$, the function is a rectangular hyperbola, meaning the asymptotes are perpendicular

Focus, Directrix & Eccentricity

A focus/foci is a point or set of points around which a curve is guided A Directrix is a fixed line perpendicular to the (major) axis of a function, which is determined by its focus and curvature Eccentricity is a function's deviation from being circular, and a constant ratio given by $e=c/a$; $$\text{eccentricity}=\frac{\text{distance from any point to the focus}}{\text{distance from any point to the directrix}}$$

• Two conic sections are congruent if the eccentricity of each are equal
• Circle eccentricities are always 0
• Ellipse eccentricities are always between 0 and 1
• Parabola eccentricities are always 1
• Hyperbola eccentricities are always between 1 and ∞
• Line eccentricities are always ∞

Degenerate Conics

Conics when either a 2D plane intercepts the vertex of a double cone, or the result of the general equation yields a non-function by real algebraic definition

No result	$A⋅x^2+A⋅y^2+1=0$
Point	$A⋅x^2+A⋅y^2=0$
Line	$D⋅x+E⋅y+F=0$
Intersecting lines	$x^2-y^2=0$
Parallel lines	$x^2-1=0$

3.5.2 Circles

Closed curves with all points equidistant to an internal point

Circumference	$2\cdot\pi\cdot r$
Area	$\pi\cdot r^2$
Arc length	$\theta\cdot r$
Sector area	$\theta\cdot r^2/2$
Chord length (k)	$2\cdot r\cdot \sin\Big(\frac{\theta}{2}\Big)$
Segment area	$\frac{r^2}{2}\cdot\Big(\theta-2\cdot\sin\Big(\frac{\theta}{2}\Big)\cdot\cos\Big(\frac{\theta}{2}\Big)\Big)$
Conic general equation	$A\cdot(x^2+y^2)+D\cdot x+E\cdot y+F=0,A≠0$
Standard equation	${(x-x_\circ)}^2+{(y-y_\circ)}^2=r^2$
Conic-standard conversions	$x_\circ=-D/(2\cdot A)$ $y_\circ=-E/(2\cdot A)$ $r^2=(D^2+E^2-4\cdot A\cdot F)/(4\cdot A^2)$
Focus Coordinates	$(x_\circ,y_\circ)$
Eccentricity	$0$
Directrix	None

Annulus

A ring formed by two concentric circles in which all features of circles apply with respect to differences involving two radii

Perimeter	$2\cdot\pi\cdot(R+r)$
Area	$\pi\cdot(R^2-r^2)$
Sector Area	$\theta\cdot(R^2-r^2)/2$

Deductive Logic for Area

Apply the area function of regular polygons, using the circumference as the perimeter and the radius as the apothem $$A=(2\cdot\pi\cdot r)\cdot\frac{r}{2}$$

Proof of Arc Length

The arc length is a fraction of the circumference, therefore can be found by the ratio to it and its angle $$\frac{a}{2\cdot\pi\cdot r}=\frac{\theta}{2\cdot\pi}$$

Proof of Sector Area

The sector area is a fraction of the circle area, therefore can be found by the ratio to it and its angle $$\frac{A_S}{\pi\cdot r^2}=\frac{\theta}{2\cdot\pi}$$

Proof of Chord Length

Use the radius and half the chord length to form a right triangle Use the sine function for the angle $$\sin\Big(\frac{\theta}{2}\Big)=\frac{k}{2\cdot r}$$ Multiply by $2\cdot r$

Proof of Segment Area

The segment area is the sector area minus the triangular area between the center and chord $$A_S=\frac{\theta\cdot r^2}{2}-A_t$$ For the triangular area, use the chord length equation for the base and right angle definition with respect to the radius to find the height

$$k=2\cdot r\cdot\sin\Big(\frac{\theta}{2}\Big)$$ $$h=r\cdot\cos\Big(\frac{\theta}{2}\Big)$$

Substitute the triangular area using these values for $(k\cdot h)/2$ $$A_S=\frac{\theta\cdot r^2}{2}-\frac{r^2}{2}\cdot 2\cdot\sin\Big(\frac{\theta}{2}\Big)\cdot\cos\Big(\frac{\theta}{2}\Big)$$ Factor $$A_S=\frac{r^2}{2}\cdot\Big(\theta-2\cdot\sin\Big(\frac{\theta}{2}\Big)\cdot\cos\Big(\frac{\theta}{2}\Big)\Big)$$

Conic-Standard Conversion

Given the conic general equation with the properties for a circle, group the $x$ terms and $y$ terms, and isolate the constant $$A\cdot x^2+D\cdot x+A\cdot y^2+E\cdot y=-F$$ Divide by $A$ $$x^2+\frac{D}{A}\cdot x+y^2+\frac{E}{A}\cdot y=-\frac{F}{A}$$ Complete the square for the $x$ and $y$ terms $$x^2+\frac{D}{A}\cdot x+\frac{D^2}{4\cdot A^2}+y^2+\frac{E}{A}\cdot y+\frac{E^2}{4\cdot A^2}=-\frac{F}{A}+\frac{D^2}{4\cdot A^2}+\frac{E^2}{4\cdot A^2}$$ Factor $${\Big(x+\frac{D}{2\cdot A}\Big)}^2+{\Big(y+\frac{E}{2\cdot A}\Big)}^2=\frac{D^2+E^2-4\cdot A\cdot F}{4\cdot A^2}$$ Equate the coefficients to the standard equation

3.5.3 Ellipses

Closed ovular curves whose points are the result of a constant sum between two internal points

Perimeter	$4\cdot a\lt 2\cdot\pi\cdot a, a\gt b$
Area	$\pi\cdot a\cdot b$
Conic General Equation	$A\cdot x^2+C\cdot y^2+D\cdot x+E\cdot y+F=0,A\cdot C>0$
Standard Equation	${(x-x_\circ)}^2/a^2+{(y-y_\circ)}^2/b^2=1$
Conic-Standard Conversions	$A=b^2$ $C=a^2$ $D=-2\cdot b^2\cdot x_\circ$ $E=-2\cdot a^2\cdot y_\circ$ $F=b^2\cdot{x_\circ}^2+a^2\cdot{y_\circ}^2-a^2\cdot b^2$ $a^2=C$ $b^2=A$ $x_\circ=-D/(2\cdot A)$ $y_\circ=-E/(2\cdot C)$
Orientation	horizontal if $C>A \land a>b$ vertical if $A>C \land b>a$
Center Coordinates	$(x_\circ,y_\circ)$
Foci Coordinates	$f=\big(x_\circ\pm\sqrt{a^2-b^2}, y_\circ\big),a>b$ $f=\left(x_\circ,y_\circ\pm \sqrt{b^2-a^2}\right),b>a$
Eccentricity	$e=f/a=\sqrt{1-b^2/a^2},a>b$ $e=f/b=\sqrt{1-a^2/b^2},b>a$
Directrix	$x=\pm a/e,a>b$ $y=\pm b/e,b>a$

Conic-Standard Conversion

Given the standard equation $$\frac{{(x-x_\circ)}^2}{a^2}+\frac{{(y-y_\circ)}^2}{b^2}=1$$ Multiply by $a^2\cdot b^2$ $$b^2\cdot{(x-x_\circ)}^2+a^2\cdot{(y-y_\circ)}^2=a^2\cdot b^2$$ Expand $$b^2\cdot x^2-2\cdot b^2\cdot x\cdot x_\circ+b^2\cdot x_\circ^2+a^2\cdot y^2-2\cdot a^2\cdot y\cdot y_\circ+a^2\cdot y_\circ^2=a^2\cdot b^2$$ Rearrange to appear as the conic general equation $$b^2\cdot x^2+a^2\cdot y^2-2\cdot b^2\cdot x_\circ\cdot x-2\cdot a^2\cdot y_\circ\cdot y+b^2\cdot x_\circ^2+a^2\cdot y_\circ^2-a^2\cdot b^2=0$$ Equate the coefficients to the conic equation

3.5.4 Parabolas

Open mirrored curves whose points are the same distance between a common internal point and an exterenal line.

Conic General Equations	vertical: $A\cdot x^2+D\cdot x+E\cdot y+F=0$ horizontal: $C\cdot y^2+D\cdot x+E\cdot y+F=0$
The following are in vertical form
Standard Equation	$y=a\cdot x^2+b\cdot x+y_\circ$
Vertex Equation	$y=a\cdot(x-x_\circ)^2+y_\circ$
Intercept Equation	$y=a\cdot(x-x_1)(x-x_2)$
Discriminant	$\Delta=b^2-4\cdot a\cdot y_\circ$ If $Δ \lt 0$, two $x$-intercepts If $Δ=0$, one $x$-intercept If $Δ>0$, no $x$-intercepts
$x$ Relationships	$x_1+x_2=-b/a$ $x_1-x_2=\sqrt{\Delta}/a$ $x_1\cdot x_2=y_\circ/a$
Conic-Standard Conversion	$a=-A/E$ $b=-D/E$ $y_\circ=-F/E$
Vertex Coordinates	$(x_\circ,y_\circ)=\big(-\frac{b}{2\cdot a},y_\circ-\frac{b^2}{4\cdot a}\big)$
$x$-Intercepts	$\lbrace (x_1,0),(x_2,0) \rbrace = \big(\frac{-b \pm \sqrt{b^2-4\cdot a\cdot y_\circ}}{2\cdot a},0\big)$
Fucus Length from Vertex	$f=\frac{1}{4\cdot a}$
Focus Coordinates	$\big(-\frac{b}{2\cdot a},y_\circ-\frac{b^2+1}{4\cdot a}\big)$
Eccentricity	$1$
Drectrix	$y=-f$

Conic-Standard Conversion

Rearrange the conic general equation to isolate the $y$ term $$E\cdot y=-A\cdot x^2-D\cdot x-F$$ Divide by $E$ $$y=-\frac{A}{E}\cdot x^2-\frac{D}{E}\cdot x-\frac{F}{E}$$ Equate the coefficients to the standard equation

Standard-Vertex Conversion

Given the standard equation, isolate the $x$ terms $$y-y_\circ=a\cdot x^2+b\cdot x$$ Divide by $a$ $$\frac{y-y_\circ}{a}=x^2+\frac{b\cdot x}{a}$$ Complete the square $$\frac{y-y_\circ}{a}+\frac{b^2}{4\cdot a^2}=x^2+\frac{b\cdot x}{a}+\frac{b^2}{4\cdot a^2}$$ Factor $$\frac{y-y_\circ}{a}+\frac{b^2}{4\cdot a^2}={\bigg(x+\frac{b}{2\cdot a}\bigg)}^2$$ Multiply by $a$ $$y-y_\circ+\frac{b^2}{4\cdot a}=a\cdot{\bigg(x+\frac{b}{2\cdot a}\bigg)}^2$$ Isolate $y$ $$y=a\cdot{\bigg(x+\frac{b}{2\cdot a}\bigg)}^2-\frac{b^2}{4\cdot a}+y_\circ$$ Equate the coefficients to the vertex equation

Standard-Intercept Conversion

Given the quadratic formula with $y=0$, find the zeros of $x$

3.5.5 Hyperbolas

A mirrored set of open mirrored curvers whose points are the difference between two common internal points.

Conic General Equation	$A\cdot x^2+C\cdot y^2+D\cdot x+E\cdot y+F=0, A\cdot C \lt 0$
Standard Equation	$(x-x_\circ)^2/a^2+(y-y_\circ)^2/b^2=1, a^2\cdot b^2 \lt 0$
Standard Equation (Real Terms Only)	$\pm (x-x_\circ)^2/a^2 \mp (y-y_\circ)^2/b^2=1$
Conic-Standard Conversion	$A=b^2$ $C=a^2$ $D=-2\cdot b^2\cdot x_\circ$ $E=-2\cdot a^2\cdot y_\circ$ $F=b^2\cdot{x_\circ}^2+a^2\cdot{y_\circ}^2-a^2\cdot b^2$ $a^2=C$ $b^2=A$ $x_\circ=-D/(2\cdot A)$ $y_\circ=-E/(2\cdot C)$
Orientation	horizontal if $C \lt 0; a \isin ℂ$ vertical if $A \lt 0; b \isin ℂ$
Vertices	$(x_\circ\pm a,y_\circ) \lor (x_\circ,y_\circ\pm b)$
Focus Coordinates	$(x_\circ\pm \sqrt{a^2+b^2},y_\circ) \lor (x_\circ,y_\circ \pm \sqrt{a^2+b^2})$
Eccentricity	$e=f/a=\sqrt{1+b^2/a^2}$
Directrix	$x=\pm a^2/f \lor y=\pm b^2/f$
Asymptotes	$y= \pm b\cdot(x-x_\circ)/a+y_\circ$

Conic-Standard Conversion

The proof is the same for ellipses, however since by definition $A\cdot C \lt 0$, either $a$ or $b$ must be imaginary. For the real-numbers-only representation, one of the terms is negated for $a$ and $b$ to both be positive.

3.5.6 Rotations 🔧

Conic section rotations occur exclusively when in the general equation $B \neq 0$ $$A\cdot x^2+B\cdot x\cdot y+C\cdot y^2+D\cdot x+E\cdot y+F=0$$ The angle of rotation is used to identify a new axis The original components are used to determine coordinates with respect to the angle of rotation



$$x=x'\cdot\cos(\theta)-y'\cdot\sin(\theta)$$ $$y=x'\cdot\sin(\theta)+y'\cdot\cos(\theta)$$ $$A\cdot\big( \big)^2+B\cdot\big( \big)\cdot\big( \big)+C\cdot\big( \big)^2+D\cdot\big( \big)+E\cdot\big( \big)+F=0$$