Add comprehensive ellipse documentation

.github/copilot-instructions.md

@@ -298,9 +298,34 @@ Documentation in `docs/src/` explains general math concepts (not code). Follow t
 
 **Mathematical Notation:**
 - Use LaTeX: `$$` for display equations, inline with `$...$`
+- **CRITICAL**: `$$` display blocks in Documenter.jl must have `$$` directly attached to content (no blank lines or return after opening or before closing)
+  - ✅ **CORRECT** (multi-line aligned block):
+    ```
+    $$\begin{aligned}
+    b &= a\sqrt{1-e^2} \\
+    c &= ae \\
+    a^2 &= b^2 + c^2
+    \end{aligned}$$
+    ```
+  - ✅ **CORRECT** (single equation): `$$equation$$`
+  - ❌ **WRONG** (blank lines or return between `$$` and content):
+    ```
+    $$
+
+    content
+
+    $$
+    ```
+- **Bullet point style**: List items starting with `-` must begin with text, then LaTeX math
+  - ✅ **CORRECT**: `- The semi-major axis is $a$`
+  - ❌ **WRONG**: `- $a$ is the semi-major axis`
+- **Equation placement style**: Prefer inline equations with "where:" at end of sentence, followed by list
+  - ✅ **CORRECT**: `For an ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where:`
+  - ❌ **AVOID**: Blank line, then equation block, then blank line, then "where:"
 - LaTeX syntax for symbols: `^\circ` not `°`, `\frac{}{}` for fractions
 - Label variables clearly: "where: $r$ = radius, $θ$ = angle"
 - Use aligned equations: `\begin{aligned}...\end{aligned}` for multi-step derivations
+- Use square brackets `[x, y]` for point coordinates consistently across documentation
 
 **MathWorld Links:**
 - Link every new mathematical term on first mention

Project.toml

@@ -1,6 +1,6 @@
 name = "Math_Foundations"
-authors = ["Aron T"]
 version = "0.1.0"
+authors = ["Aron T"]
 
 [deps]
 AMRVW = "233ec0c9-6b8d-4766-b9a2-a16e389fc38a"
@@ -10,6 +10,7 @@ Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 DrWatson = "634d3b9d-ee7a-5ddf-bec9-22491ea816e1"
 FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
 GLMakie = "e9467ef8-e4e7-5192-8a1a-b1aee30e663a"
+GeometryBasics = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
 IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a"
 ImageShow = "4e3cecfd-b093-5904-9786-8bbb286a6a31"
 LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"

docs/make.jl

@@ -22,7 +22,10 @@ makedocs(;
             "Geometry/02 Quadrilaterals.md",
             "Geometry/03 Polygons.md",
             "Geometry/04 Circles.md",
-            # "Geometry/04 Hyperbola.md"  # TODO: Uncomment when ready to show
+            "Geometry/05 Ellipses.md",
+        #   "Geometry/06 Hyperbola.md" TODO: Uncomment when ready to show,
+        #   "Geometry/07 Parabola.md" TODO: Uncomment when ready to show
+
         ],
         # "Linear Algebra" => [  # TODO: Uncomment when ready to show
         #     "Linear Algebra/01 Systems of Equations.md",

docs/src/Geometry/05 Ellipses.md

@@ -0,0 +1,324 @@
+# Ellipses
+
+An [ellipse](https://mathworld.wolfram.com/Ellipse.html) is a plane figure that like a circle, also has curvilinear edges, but is more general. Ellipses are fundamental [conic sections](https://mathworld.wolfram.com/ConicSection.html) with applications in orbital mechanics, optics, engineering, and architecture.
+
+## Definition and Basic Properties
+
+### Set/Geometric Definition
+
+An ellipse can be defined as the set of all points $P$ (the [locus](https://mathworld.wolfram.com/Locus.html)) in a plane such that the sum of the distances to two fixed points $F_1$ and $F_2$ (the foci--see below) is constant: $$\{P : |PF_1| + |PF_2| = 2a\}$$ where:
+
+- The distance from $P$ to $F_1$ is $|PF_1|$
+- The distance from $P$ to $F_2$ is $|PF_2|$
+- The constant sum $2a$ is the length of the major axis (see below) of the ellipse
+- The semi-major axis (see below) is $a$ (half the major axis length)
+
+### Parts of an Ellipse
+
+- **[Foci](https://mathworld.wolfram.com/Focus.html):** Two fixed points $F_1$ and $F_2$ inside the ellipse such that the sum of distances to any point on the ellipse is constant
+- **Center:** The midpoint between the two foci
+- **Major Axis:** The longest diameter of the ellipse, passing through both foci, with length $2r$ or $2a$
+- **Minor Axis:** The shortest diameter, perpendicular to the major axis at the center, with length $2b$
+- **Semi-Major Axis:** Half the length of the major axis, denoted $a$
+- **Semi-Minor Axis:** Half the length of the minor axis, denoted $b$
+- **Vertices:** The endpoints of the major axis, located at a distance $a$ from the center
+- **Co-Vertices:** The endpoints of the minor axis, located at a distance $b$ from the center
+
+### Standard Form Equation
+
+#### Derivation of Standard Form
+
+Let's derive the standard form of the ellipse equation using the Pythagorean theorem and the distance definition. An ellipse is formed from the set of all points $[x, y]$ for which the sum $r_1 + r_2$ of the distances to two fixed points $A = [a_1, a_2]$ and $B = [b_1, b_2]$ (called focal points) has a constant value $2r$. Using the Pythagorean theorem we get:
+
+$$\begin{aligned}
+r_1 &= \sqrt{(x - a_1)^2 + (y - a_2)^2} \\
+r_2 &= \sqrt{(x - b_1)^2 + (y - b_2)^2} \\
+r_1 + r_2 &= 2r
+\end{aligned}$$
+
+For the sake of simplicity, assume that the two foci lie on the x-axis and that the center of the ellipse (center of the foci) is the origin $[0, 0]$: $A = [-c, 0]$ and $B = [c, 0]$, where $c > 0$. Substituting these values in the equation above we get:
+
+$$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2r (or 2a)$$
+
+In this case the two _major vertices_ of the ellipse are the points $H_1 = [-r, 0]$ and $H_2 = [r, 0]$, the line of length $2r$ (or $2a$) connecting these points is called the _major axis_ of the ellipse $r$ (or $a$) is called the _major axis radius_. The two _minor vertices_ of the ellipse are then the points of the form $N_1 = [0, b]$ and $N_2 = [0, -b]$ on the ellipse-–that is, the points of intersection of such an ellipse with the y-axis.
+
+The points $N_1$ and $N_2$ are also the intersections of the circles of radius $r$ about either of the foci. Hence we can obtain from the Pythagorean theorem:
+
+$$\begin{aligned}
+b^2 &= r^2 - c^2 \\
+b &= \sqrt{r^2 - c^2} \\
+\end{aligned}$$
+
+Thus point $N_1 = [0, \sqrt{r^2 - c^2}]$ and $N_2 = [0, -\sqrt{r^2 - c^2}]$. The minor vertices form the minor axis of the ellipse and $b = \sqrt{r^2 - c^2}$ denotes the minor axis radius.
+
+#### Circle as Special Case
+
+When the two foci coincide at the center ($c = 0$), the elliptic equation becomes:
+
+$$\begin{aligned}
+\sqrt{x^2 + y^2} + \sqrt{x^2 + y^2} &= 2r \text{ and then} \\
+2 \sqrt{x^2 + y^2} &= 2r \\
+\end{aligned}$$
+
+Cancelling $2$ and then squaring both sides gives  equation $x^2 + y^2 = r^2$, which is the standard equation of a circle with radius $r$ centered at the origin.
+
+### Midpoint Equation
+
+The standard form of the ellipse centered at the origin with foci on the x-axis can be simplified to: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where:
+
+- $x$ and $y$ are the coordinates $[x,y]$ of any point on the ellipse
+- The semi-major axis (assuming $a \geq b$) is $a$
+- The semi-minor axis is $b$
+- The relationship $b^2 = a^2 - c^2$ connects the semi-minor axis to the distance between foci
+- The foci are located at $[-c, 0]$ and $[c, 0]$
+- The distance between the foci is $2c$
+- The major axis length is $2a$
+- The minor axis length is $2b$
+- The center is at the origin $[0, 0]$
+- The vertices are at $[-a, 0]$ and $[a, 0]$
+- The co-vertices are at $[0, -b]$ and $[0, b]$
+- If the major axis is vertical instead of horizontal, the equation is: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with foci at $[0, -c]$ and $[0, c]$
+
+Some additional notes:
+
+- From the triangle inequality it follows that the distance between the foci ($2c$) must be smaller than $r_1 + r_2 = 2r$. This criterion shows how large we should choose the major axis radius $r$ at least in order to be able to construct an ellipse with given foci.
+- If points are given on an ellipse to be constructed, unlike a  circle, three points are generally not enough to define an ellipse. Four suitable points are needed.
+- Ellipses are symmetrical figures with respect to the major and minor axes.
+
+The full (and quite complex) derivation of the midpoint equation is provided in the Appendix at the end of this document.
+
+### Translated Ellipse
+
+For an ellipse centered at point $[h, k]$: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
+
+## Key Parameters of an Ellipse
+
+### Eccentricity
+
+The [eccentricity](https://mathworld.wolfram.com/Eccentricity.html) measures how "stretched" the ellipse is: $e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}$ where $0 \leq e < 1$:
+
+- A circle (special case) has $e = 0$
+- An increasingly elongated ellipse has $e \to 1$
+
+### Foci
+
+The foci are located at distance $c$ from the center along the major axis: $c = ae = \sqrt{a^2 - b^2}$
+
+For a horizontal major axis centered at origin: $F_1 = [-c, 0]$ and $F_2 = [c, 0]$. For a vertical major axis centered at origin: $F_1 = [0, -c]$ and $F_2 = [0, c]$
+
+### Relationship Between Parameters
+
+$$\begin{aligned}
+b &= a\sqrt{1-e^2} \\
+c &= ae \\
+a^2 &= b^2 + c^2
+a = \sqrt{b^2 + c^2}
+\end{aligned}$$
+
+## Parametric Form of an Ellipse
+
+Ellipses can be expressed parametrically, similar to circles but with different scalings:
+
+$$\begin{aligned}
+x(t) &= a\cos(t) \\
+y(t) &= b\sin(t)
+\end{aligned}$$ where $t \in (0, 2\pi)$ is the parameter.
+
+For a translated ellipse centered at $[h, k]$:
+
+$$\begin{aligned}
+x(t) &= h + a\cos(t) \\
+(t) &= k + b\sin(t)
+\end{aligned}$$
+
+## Polar Form
+
+The most common form uses polar coordinates with the origin at one of the foci: $$r(\theta) = \frac{a(1 - e^2)}{1 + e\cos(\theta)}$$ where $r$ is the distance from the chosen focus and $\theta$ is the angle from the major axis.
+
+This form is particularly useful in orbital mechanics, where celestial bodies follow elliptical orbits with the central body at one focus (Kepler's First Law).
+
+There is also a polar form with the center at the origin, but it is less commonly used: $$r(\theta) = \frac{ab}{\sqrt{(b\cos(\theta))^2 + (a\sin(\theta))^2}}$$
+
+## Geometric Properties
+
+### Area
+
+The area enclosed by an ellipse is: $$F = \pi ab$$
+
+This can be seen as a scaled version of a circle's area, where the circle of radius $a$ is compressed by factor $b/a$ in one direction.
+
+### Perimeter
+
+The perimeter (circumference) of an ellipse cannot be expressed in elementary functions and requires us of integral calculus and knowledge of elliptical integrals](https://mathworld.wolfram.com/EllipticIntegral.html). This is outside the scope of this document.
+
+There are, however, several useful approximations for the perimeter:
+
+**Ramanujan's Approximation** (very accurate): $$p \approx \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right)$$ where $h = \frac{(a-b)^2}{(a+b)^2}$
+
+**Simpler Approximation**: $$p \approx \pi\sqrt{2(a^2 + b^2)}$$
+
+**Another Approximation**: $$p \approx \pi \left(1.5 (a+b) - \sqrt{ab}\right)$$
+
+**Ramanujan's Second Form**: $$p \approx \pi\left(3(a+b) - \sqrt{(a+3b)(3a+b)}\right)$$
+
+### Curvature
+
+The curvature at parameter $t$ is: $$\kappa(t) = \frac{ab}{(b^2\cos^2 t + a^2\sin^2 t)^{3/2}}$$
+
+Maximum curvature occurs at the ends of the minor axis: $\kappa_{\text{max}} = \frac{a}{b^2}$
+
+Minimum curvature occurs at the ends of the major axis: $\kappa_{\text{min}} = \frac{b}{a^2}$
+
+## Directrix
+
+Each focus has an associated [directrix](https://mathworld.wolfram.com/ConicSectionDirectrix.html), a line perpendicular to the major axis. The ratio of distances from any point on the ellipse to a focus and its corresponding directrix equals the eccentricity: $$\frac{a}{d} = e$$
+
+The directrix is located at distance $\frac{a^2}{c} = \frac{a}{e}$ from the center.
+
+## Reflective Property
+
+A ray of light emanating from one focus will reflect off the ellipse and pass through the other focus. This property is used in:
+
+- **Whispering galleries**: Sound waves from one focus are audible at the other
+- **Elliptical mirrors**: Used in certain telescope designs
+- **Medical applications**: Lithotripsy uses elliptical reflectors to focus shock waves on kidney stones
+
+## Special Cases and Relationships
+
+### Circle as Special Case
+
+When $a = b$, the ellipse becomes a circle:
+
+- **Eccentricity** is $e = 0$
+- **Both foci** coincide at the center
+- **All radii** are equal
+
+### Rotation to 3D
+
+- **Oblate spheroid**: Ellipse rotated about its minor axis (like Earth's shape)
+- **Prolate spheroid**: Ellipse rotated about its major axis (like a rugby ball)
+
+### Relationship to Other Conics
+
+Ellipses are one of the four types of [conic sections](https://mathworld.wolfram.com/ConicSection.html):
+
+- **Circle:** Special case of ellipse with $e = 0$
+- **Parabola:** Limiting case as $e \to 1$
+- **Hyperbola:** When $e > 1$
+
+## Applications
+
+### Astronomy and Orbital Mechanics
+
+- Planetary orbits are elliptical (Kepler's First Law)
+- The Sun is at one focus of each planet's orbital ellipse
+- Satellites, comets, and asteroids follow elliptical paths
+
+### Engineering and Architecture
+
+- Elliptical gears can rotate smoothly against each other
+- Whispering galleries in architecture
+- Elliptical arches distribute load efficiently
+
+### Optics
+
+- Elliptical mirrors focus light from one focus to another
+- Used in certain telescope and reflector designs
+
+### Nature
+
+- Orbits of electrons in some atomic models (historically)
+- Cross-sections of many biological structures
+- Shadows cast by circles viewed at an angle
+
+## Construction Methods
+
+### String Method
+
+The classical method using the two-focus definition:
+
+1. Pin a string at both foci with total length $2a$
+2. Keep the string taut with a pencil
+3. Move the pencil to trace the ellipse
+
+### Trammel Construction
+
+If endpoints of a segment move along two perpendicular lines, a fixed point on the segment traces an ellipse.
+
+## Historical Notes
+
+- **Menaechmus** (ca. 350 BCE): First studied ellipses
+- **Apollonius** (ca. 200 BCE): Named the ellipse and studied its properties
+- **Johannes Kepler** (1609): Discovered that planetary orbits are elliptical
+- **Edmund Halley** (1705): Showed that Halley's Comet follows an elliptical orbit
+
+## See Also
+
+- [Circles](04 Circles.md) - Special case of ellipse
+- [Conic Sections](https://mathworld.wolfram.com/ConicSection.html)
+
+## Appendix: Derivation of Standard Form Equation
+
+By squaring and rearranging the standard form for the ellipse, we can derive the simplified midpoint equation. Starting from the standard form:
+
+$$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2r$$
+
+Isolate one of the square roots:
+
+$$\sqrt{(x + c)^2 + y^2} = 2r - \sqrt{(x - c)^2 + y^2}$$
+
+Now square both sides:
+
+$$(x + c)^2 + y^2 = (2r - \sqrt{(x - c)^2 + y^2})^2$$
+
+Expanding the right side:
+
+$$(x + c)^2 + y^2 = 4r^2 - 4r\sqrt{(x - c)^2 + y^2} + (x - c)^2 + y^2$$
+
+Cancel $y^2$ from both sides and rearrange:
+
+$$4r\sqrt{(x - c)^2 + y^2} = 4r^2 + (x - c)^2 - (x + c)^2$$
+
+Simplifying the right side:
+
+$$4r\sqrt{(x - c)^2 + y^2} = 4r^2 - 4cx$$
+
+Divide both sides by $4r$:
+
+$$\sqrt{(x - c)^2 + y^2} = r - \frac{cx}{r}$$
+
+Now square both sides again:
+
+$$(x - c)^2 + y^2 = \left(r - \frac{cx}{r}\right)^2$$
+
+Expanding the right side:
+
+$$(x - c)^2 + y^2 = r^2 - 2cx + \frac{c^2x^2}{r^2}$$
+
+Rearranging gives:
+
+$$y^2 = r^2 - 2cx + \frac{c^2x^2}{r^2} - (x - c)^2$$
+
+Expanding $(x - c)^2$:
+
+$$y^2 = r^2 - 2cx + \frac{c^2x^2}{r^2} - (x^2 - 2cx + c^2)$$
+
+Simplifying:
+
+$$y^2 = r^2 - x^2 + \frac{c^2x^2}{r^2} - c^2$$
+
+Combining like terms:
+
+$$y^2 = (r^2 - c^2) - x^2\left(1 - \frac{c^2}{r^2}\right)$$
+
+Factoring:
+
+$$y^2 = (r^2 - c^2) - x^2\frac{r^2 - c^2}{r^2}$$
+
+Dividing both sides by $(r^2 - c^2)$:
+
+$$\frac{y^2}{r^2 - c^2} + \frac{x^2}{r^2} = 1$$
+
+Finally,since  $b^2 = r^2 - c^2$, we have the simplified midpoint equation form of the ellipse equation (using the more standard $a$ for semi-major axis and $b$ for semi-minor axis):
+
+$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

docs/src/Geometry/07 Parabola.md

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+# Parabola