From f7fd3e8a7e56d527e7ab0f826999515a1f6e7825 Mon Sep 17 00:00:00 2001
From: Chris Proctor <chris@chrisproctor.net>
Date: Mon, 28 Jul 2025 13:29:18 -0400
Subject: [PATCH] Initial commit

---
 .commit_template        |  11 +
 .gitignore              |   1 +
 geometry/base.py        |  19 ++
 geometry/drawing.py     |  34 +++
 geometry/point.py       |  73 +++++
 geometry/validation.py  |  31 +++
 geometry/vector.py      | 141 ++++++++++
 poetry.lock             | 593 ++++++++++++++++++++++++++++++++++++++++
 pyproject.toml          |  28 ++
 sim_billiards.py        |  38 +++
 sim_solar_system.py     |  15 +
 simulation/__init__.py  |  60 ++++
 simulation/ball.py      |  37 +++
 simulation/collision.py |  39 +++
 simulation/drawing.py   |  31 +++
 simulation/gravity.py   |  13 +
 tests/test_point.py     |  48 ++++
 tests/test_vector.py    | 101 +++++++
 18 files changed, 1313 insertions(+)
 create mode 100644 .commit_template
 create mode 100644 .gitignore
 create mode 100644 geometry/base.py
 create mode 100644 geometry/drawing.py
 create mode 100644 geometry/point.py
 create mode 100644 geometry/validation.py
 create mode 100644 geometry/vector.py
 create mode 100644 poetry.lock
 create mode 100644 pyproject.toml
 create mode 100644 sim_billiards.py
 create mode 100644 sim_solar_system.py
 create mode 100644 simulation/__init__.py
 create mode 100644 simulation/ball.py
 create mode 100644 simulation/collision.py
 create mode 100644 simulation/drawing.py
 create mode 100644 simulation/gravity.py
 create mode 100644 tests/test_point.py
 create mode 100644 tests/test_vector.py

diff --git a/.commit_template b/.commit_template
new file mode 100644
index 0000000..bf56e73
--- /dev/null
+++ b/.commit_template
@@ -0,0 +1,11 @@
+
+
+# -----------------------------------------------------------------
+# Write your commit message above this line.
+#
+# The first line should be a quick description of what you changed.
+# Then leave a blank line. 
+# Then write a few sentences reflecting on one moment from your last
+# work session. What were you feeling? If you solved a problem, 
+# what strategy helped you?
+
diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..0d20b64
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1 @@
+*.pyc
diff --git a/geometry/base.py b/geometry/base.py
new file mode 100644
index 0000000..4c6f7e9
--- /dev/null
+++ b/geometry/base.py
@@ -0,0 +1,19 @@
+from caseconverter import snakecase
+
+class GeometryObject:
+    """A base class for geometry objects.
+    This base class provides some consistent behavior for all geometry objects.
+
+    Note: This class uses some advanced techniques which you aren't expected to
+    understand. If you're interested, by all means keep reading, and ask a teacher
+    if you have questions. 
+    """
+
+    def __new__(cls, value):
+        """Overrides default constructor to support construction from other objects.
+        When value is an instance of CustomClass, checks cls for a `from_custom_class` 
+        method which will be called to produce a new instance. 
+        """
+        method_name = value.__class__.__name__
+
+
diff --git a/geometry/drawing.py b/geometry/drawing.py
new file mode 100644
index 0000000..ea0c25d
--- /dev/null
+++ b/geometry/drawing.py
@@ -0,0 +1,34 @@
+from turtle import *
+from superturtle.movement import fly
+
+def centered_circle(radius):
+    """Draws a circle, centered on the turtle's current position.
+    The built-in circle function annoyingly draws a circle whose center
+    is one radius left of the turtle.
+    """
+    right(90)
+    penup()
+    forward(radius)
+    pendown()
+    left(90)
+    circle(radius)
+    left(90)
+    penup()
+    forward(radius)
+    pendown()
+    right(90)
+
+def write_label(point, label, align=None, font=None):
+    fly(point.x, point.y)
+    write(label, align=align, font=font)
+
+def line_end_arrow():
+    "Draws an arrow"
+    left(20)
+    back(20)
+    forward(20)
+    right(40)
+    back(20)
+    forward(20)
+    left(20)
+
diff --git a/geometry/point.py b/geometry/point.py
new file mode 100644
index 0000000..7448556
--- /dev/null
+++ b/geometry/point.py
@@ -0,0 +1,73 @@
+from geometry.validation import is_coordinates, expect_type
+from geometry.drawing import centered_circle, write_label
+from superturtle.movement import fly
+from turtle import begin_fill, end_fill, fillcolor
+
+class Point:
+    """A Point is a location in 2D space, identified by its
+    x and y coordinates. A Point can be initialized in multiple ways:
+
+    - With no value: The Point will be the origin (0, 0).
+    - With an ordered pair of numbers: The Point will have those coordinates.
+    - With a Vector: The Point will have the Vector's x and y coordinates.
+    """
+
+    def __init__(self, value=None):
+        from geometry.vector import Vector
+        if value is None:
+            self.x, self.y = (0, 0)
+        elif is_coordinates(value):
+            self.x, self.y = value
+        elif isinstance(value, Vector):
+            self.x, self.y = value.x, value.y
+        else:
+            raise TypeError(f"Can't create a Point from {value}.")
+
+    def draw(self, label=None):
+        fly(self.x, self.y)
+        fillcolor("black")
+        begin_fill()
+        centered_circle(2)
+        end_fill()
+        if label:
+            write_label(self, label)
+
+    def __str__(self):
+        return f"({self.x}, {self.y})"
+
+    def __repr__(self):
+        return f"Point(({self.x}, {self.y}))"
+
+    def __eq__(self, other):
+        """Implements equality checking with a == b.
+        Two Points are equal if their coordinates are the same.
+        A Point can only be equal to another Point.
+        """
+        try:
+            expect_type(other, Point)
+            return (self.x, self.y) == (other.x, other.y)
+        except TypeError:
+            return False
+
+    def __add__(self, other):
+        """Implements addition with a + b. Other can be:
+        - a Vector: interpreted as moving the point in the direction of the Vector, 
+          returning a new Point.
+        """
+        from geometry.vector import Vector
+        expect_type(other, Vector)
+        return Point((self.x + other.x, self.y + other.y))
+
+    def __sub__(self, other):
+        """Implements subtraction with a - b. Other can be:
+        - a Point: interpreted as "what's needed to get from here to there,
+          returning a Vector.
+        - a Vector:interpreted as moving the point in the (negative) direction 
+          of the vector and returns a Point.
+        """
+        from geometry.vector import Vector
+        expect_type(other, (Point, Vector))
+        if isinstance(other, Point):
+            return Vector((self.x - other.x, self.y - other.y))
+        elif isinstance(other, Vector):
+            return Point((self.x - other.x, self.y - other.y))
diff --git a/geometry/validation.py b/geometry/validation.py
new file mode 100644
index 0000000..0deb343
--- /dev/null
+++ b/geometry/validation.py
@@ -0,0 +1,31 @@
+#
+#
+#
+
+def is_numeric(value):
+    "Checks whether value is an integer or a float (decimal number)."
+    return isinstance(value, int) or isinstance(value, float)
+
+def is_coordinates(value):
+    "Checks whether value is a tuple of two numeric values."
+    try:
+        x, y = value
+        return is_numeric(x) and is_numeric(y)
+    except (TypeError, ValueError):
+        return False
+
+def has_method(obj, method_name):
+    "Checks whether the object has a method called `method_name`."
+    attr = getattr(obj, method_name, None)
+    return callable(attr)
+
+def expect_type(obj, expected_type):
+    """Checks whether obj is of the expected type. 
+    If multiple types are provided, checks whether obj is one of the types.
+    """
+    try:
+        if not any(isinstance(obj, t) for t in expected_type):
+            raise TypeError(f"Expected {obj} to be one of the following types: {expected_type}")
+    except TypeError:
+        if not isinstance(obj, expected_type):
+            raise TypeError(f"Expected {obj} to be a {expected_type}")
diff --git a/geometry/vector.py b/geometry/vector.py
new file mode 100644
index 0000000..3ab4ab9
--- /dev/null
+++ b/geometry/vector.py
@@ -0,0 +1,141 @@
+from geometry.point import Point
+from geometry.validation import is_coordinates, expect_type
+from geometry.drawing import line_end_arrow
+from math import sqrt, atan
+from superturtle.movement import fly
+from turtle import setheading, forward
+
+class Vector:
+    """A Vector represents movement in 2D space. Like a Point, a
+    Vector is defined by x and y coordinates. However, Points and Vectors
+    are different. Here are a few things you can do with Vectors which it doesn't 
+    make sense to do with Points:
+    - Multiply them by a number (multiplying a vector by 2 means twice the movement)
+    - Negate them (-v means moving in the opposite direction of v)
+    - Find their length
+
+    A Vector can be initialized in multiple ways:
+
+    - With no value: Represents no motion at all: (0, 0).
+    - With an ordered pair of numbers: The Vector will have those coordinates.
+    - With a Point: The Vector will have the Point's x and y coordinates.
+    """
+    def __init__(self, value=None):
+        if value is None:
+            self.x, self.y = (0, 0)
+        elif is_coordinates(value):
+            self.x, self.y = value
+        elif isinstance(value, Point):
+            self.x, self.y = value.x, value.y
+        else:
+            raise TypeError(f"Can't create a Vector from {value}.")
+
+    def draw(self, origin=None):
+        "Draws the vector."
+        origin = origin or Point()
+        expect_type(origin, Point)
+        fly(origin.x, origin.y)
+        θ = atan(self.y / self.x)
+        setheading(θ)
+        forward(self.mag())
+        line_end_arrow()
+
+    def __str__(self):
+        return f"〈{self.x}, {self.y}〉"
+
+    def __repr__(self):
+        return f"Vector(({self.x}, {self.y}))"
+
+    def __eq__(self, other):
+        """Implements equality checking with a == b.
+        Two Vectors are equal if their coordinates are the same.
+        A Vector can only be equal to another Vector.
+        """
+        try:
+            expect_type(other, Vector)
+            return (self.x, self.y) == (other.x, other.y)
+        except TypeError:
+            return False
+
+    def __neg__(self):
+        """Implements negation with -a.
+        """
+        return Vector((-self.x, -self.y))
+
+    def __add__(self, other):
+        """Implements addition with a + b. Other can be:
+        - a Vector: interpreted as combining the movement of self and other, 
+          returning a new Vector.
+        - a Point: interpreted as moving the point in the direction of the Vector,
+          returning a new Point.
+        """
+        expect_type(other, (Point, Vector))
+        if isinstance(other, Vector):
+            return Vector((self.x + other.x, self.y + other.y))
+        elif isinstance(other, Point):
+            return Point((self.x + other.x, self.y + other.y))
+
+    def __sub__(self, other):
+        """Implements subtraction with a - b. Other must be a Vector,
+        interpreted as subtraction the movement of other from self and
+        returning a new Vector.
+        """
+        expect_type(other, Vector)
+        return Vector((self.x - other.x, self.y - other.y))
+
+    def __mul__(self, other):
+        """Implements multiplication with a * 2, interpreted as scaling the
+        vector by other. Other must be numeric.
+        """
+        expect_type(other, (int, float))
+        return Vector((self.x * other, self.y * other))
+
+    def __rmul__(self, other):
+        """Implements multiplication with 2 * a. 
+        """
+        return self * other
+
+    def __truediv__(self, other):
+        """Implements division with a / b, interpreted as dividing the
+        vector by other. Other must be numeric.
+        """
+        expect_type(other, (int, float))
+        if other == 0:
+            raise ZeroDivisionError("Can't divide a Vector by zero.")
+        return Vector((self.x / other, self.y / other))
+
+    def __bool__(self):
+        """Implements truthiness--whether an instance of Vector is treated as
+        True or False. Any Vector other than the zero vector is True.
+        """
+        return self.mag() != 0
+
+    def mag(self):
+        """Returns the magnitude (length) of the vector, using the Pythagorean formula.
+        """
+        return sqrt(self.x * self.x + self.y * self.y)
+
+    def scale(self, mag):
+        """Returns a Vector pointing in the same direction with the given magnitude.
+        """
+        expect_type(mag, (int, float))
+        if self.mag() > 0:
+            return self * (mag / self.mag())
+        else:
+            raise ValueError("Can't scale the zero vector.")
+
+    def normalize(self):
+        """Returns a Vector pointing in the same direction with a magnitude of 1.
+        """
+        return self.scale(1)
+
+    def dot(self, other):
+        """Computes the dot product of this vector with another. 
+        """
+        expect_type(other, Vector)
+        return self.x * other.x + self.y * other.y
+
+    def project(self, other):
+        """Projects this vector onto another.
+        """
+        return other.scale(self.dot(other.normalize()))
diff --git a/poetry.lock b/poetry.lock
new file mode 100644
index 0000000..c839265
--- /dev/null
+++ b/poetry.lock
@@ -0,0 +1,593 @@
+# This file is automatically @generated by Poetry 2.1.3 and should not be changed by hand.
+
+[[package]]
+name = "alabaster"
+version = "0.7.16"
+description = "A light, configurable Sphinx theme"
+optional = false
+python-versions = ">=3.9"
+groups = ["docs"]
+files = [
+    {file = "alabaster-0.7.16-py3-none-any.whl", hash = "sha256:b46733c07dce03ae4e150330b975c75737fa60f0a7c591b6c8bf4928a28e2c92"},
+    {file = "alabaster-0.7.16.tar.gz", hash = "sha256:75a8b99c28a5dad50dd7f8ccdd447a121ddb3892da9e53d1ca5cca3106d58d65"},
+]
+
+[[package]]
+name = "babel"
+version = "2.17.0"
+description = "Internationalization utilities"
+optional = false
+python-versions = ">=3.8"
+groups = ["docs"]
+files = [
+    {file = "babel-2.17.0-py3-none-any.whl", hash = "sha256:4d0b53093fdfb4b21c92b5213dba5a1b23885afa8383709427046b21c366e5f2"},
+    {file = "babel-2.17.0.tar.gz", hash = "sha256:0c54cffb19f690cdcc52a3b50bcbf71e07a808d1c80d549f2459b9d2cf0afb9d"},
+]
+
+[package.extras]
+dev = ["backports.zoneinfo ; python_version < \"3.9\"", "freezegun (>=1.0,<2.0)", "jinja2 (>=3.0)", "pytest (>=6.0)", "pytest-cov", "pytz", "setuptools", "tzdata ; sys_platform == \"win32\""]
+
+[[package]]
+name = "certifi"
+version = "2025.6.15"
+description = "Python package for providing Mozilla's CA Bundle."
+optional = false
+python-versions = ">=3.7"
+groups = ["docs"]
+files = [
+    {file = "certifi-2025.6.15-py3-none-any.whl", hash = "sha256:2e0c7ce7cb5d8f8634ca55d2ba7e6ec2689a2fd6537d8dec1296a477a4910057"},
+    {file = "certifi-2025.6.15.tar.gz", hash = "sha256:d747aa5a8b9bbbb1bb8c22bb13e22bd1f18e9796defa16bab421f7f7a317323b"},
+]
+
+[[package]]
+name = "charset-normalizer"
+version = "3.4.2"
+description = "The Real First Universal Charset Detector. Open, modern and actively maintained alternative to Chardet."
+optional = false
+python-versions = ">=3.7"
+groups = ["docs"]
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+[package.extras]
+brotli = ["brotli (>=1.0.9) ; platform_python_implementation == \"CPython\"", "brotlicffi (>=0.8.0) ; platform_python_implementation != \"CPython\""]
+h2 = ["h2 (>=4,<5)"]
+socks = ["pysocks (>=1.5.6,!=1.5.7,<2.0)"]
+zstd = ["zstandard (>=0.18.0)"]
+
+[metadata]
+lock-version = "2.1"
+python-versions = ">=3.10, <4.0"
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diff --git a/pyproject.toml b/pyproject.toml
new file mode 100644
index 0000000..43ce908
--- /dev/null
+++ b/pyproject.toml
@@ -0,0 +1,28 @@
+[project]
+name = "problemset-geometry"
+version = "0.1.0"
+description = ""
+authors = [
+    {name = "Chris Proctor",email = "chris@chrisproctor.net"}
+]
+license = {text = "MIT"}
+requires-python = ">=3.10, <4.0"
+dependencies = [
+    "superturtle (>=1.0.0,<2.0.0)"
+]
+
+[build-system]
+requires = ["poetry-core>=2.0.0,<3.0.0"]
+build-backend = "poetry.core.masonry.api"
+
+[tool.poetry]
+packages = [
+    {include = "geometry"}
+]
+
+[tool.poetry.group.docs]
+optional = true
+
+[tool.poetry.group.docs.dependencies]
+sphinx = "^7.3.7"
+sphinx-rtd-theme = "^2.0.0"
diff --git a/sim_billiards.py b/sim_billiards.py
new file mode 100644
index 0000000..e2cf429
--- /dev/null
+++ b/sim_billiards.py
@@ -0,0 +1,38 @@
+from turtle import *
+from math import sqrt
+from geometry.validation import expect_type
+from geometry.point import Point
+from geometry.vector import Vector
+from simulation import Simulation
+from simulation.ball import Ball
+
+def get_ball_positions(origin, spacing, columns=5):
+    """Returns a list of Points in a triangle, growing to the right of position.
+    spacing is the distance between each point,
+    columns is the number of columns in the triangle, and
+    jiggle is the amount of randomness to add to each ball's position.
+    """
+    expect_type(origin, Point)
+    positions = []
+    col_spacing = spacing * sqrt(3) / 2
+    for col in range(columns):
+        offset_x = col_spacing * col
+        for row in range(col + 1):
+            offset_y = spacing * (row - col / 2)
+            offset = Vector((offset_x, offset_y))
+            positions.append(origin + offset)
+    return positions
+
+r = 10
+positions = get_ball_positions(Point((100, 0)), 2 * r + 2)
+rack = [Ball(p, radius=r, color="red") for p in positions]
+cue_ball = Ball(
+    Point((-250, 0)), 
+    radius=r, 
+    velocity=Vector((9, 0)), 
+    mass=1,
+    color="white"
+)
+
+sim = Simulation(rack + [cue_ball], e=1, boundary=250)
+sim.run(1000)
diff --git a/sim_solar_system.py b/sim_solar_system.py
new file mode 100644
index 0000000..00006f9
--- /dev/null
+++ b/sim_solar_system.py
@@ -0,0 +1,15 @@
+# A simulation of a solar system, with a sun and three planets.
+# Gravity is the only force acting on the system.
+
+from simulation import Simulation
+from simulation.ball import Ball
+from geometry.point import Point
+from geometry.vector import Vector
+
+sun = Ball(Point(), mass=100000, radius=20, color="yellow")
+a = Ball(Point((100, 0)), velocity=Vector((0, 14)), mass=1, radius=10, color="blue")
+b = Ball(Point((-80, 0)), velocity=Vector((0, -14)), mass=1, radius=10, color="red")
+c = Ball(Point((0, 200)), velocity=Vector((-5, 0)), mass=2, radius=10, color="green")
+
+sim = Simulation([sun, a, b, c], g=0.2)
+sim.run(1000)
diff --git a/simulation/__init__.py b/simulation/__init__.py
new file mode 100644
index 0000000..a58d4e1
--- /dev/null
+++ b/simulation/__init__.py
@@ -0,0 +1,60 @@
+from itertools import combinations
+from superturtle.movement import fly, no_delay
+from superturtle.animation import animate
+from simulation.gravity import get_gravity_force
+from simulation.collision import collide2d
+from simulation.drawing import draw_boundary
+
+class Simulation:
+    def __init__(self, objects, g=None, e=None, boundary=None):
+        self.objects = objects
+        self.g = g
+        self.e = e
+        self.boundary = boundary
+        self.collisions = set()
+
+    def run(self, *args):
+        for frame in animate(*args):
+            for a, b in combinations(self.objects, 2):
+                if self.g is not None:
+                    force = get_gravity_force(a, b, self.g)
+                    a.acceleration += force / a.mass
+                    b.acceleration -= force / b.mass
+                if self.e is not None:
+                    if a.collided_with(b):
+                        self.collide(a, b)
+                    else:
+                        self.clear_collision(a, b)
+            for obj in self.objects:
+                if self.boundary:
+                    draw_boundary(self.boundary)
+                    self.bounce_object_if_outside_boundary(obj)
+                obj.update()
+                fly(obj.position.x, obj.position.y)
+                obj.draw()
+
+    def collide(self, a, b):
+        if not self.currently_colliding(a, b):
+            self.collisions.add((a, b))
+            av, bv = collide2d(a, b, self.e)
+            a.velocity = av
+            b.velocity = bv
+
+    def currently_colliding(self, a, b):
+        return (a, b) in self.collisions or (b, a) in self.collisions
+
+    def clear_collision(self, a, b):
+        if (a, b) in self.collisions:
+            self.collisions.remove((a, b))
+        if (b, a) in self.collisions:
+            self.collisions.remove((b, a))
+
+    def bounce_object_if_outside_boundary(self, obj):
+        if obj.position.x + obj.radius > self.boundary:
+            obj.velocity.x = -abs(obj.velocity.x)
+        if obj.position.x - obj.radius < -self.boundary:
+            obj.velocity.x = abs(obj.velocity.x)
+        if obj.position.y + obj.radius > self.boundary:
+            obj.velocity.y = -abs(obj.velocity.y)
+        if obj.position.y - obj.radius < -self.boundary:
+            obj.velocity.y = abs(obj.velocity.y)
diff --git a/simulation/ball.py b/simulation/ball.py
new file mode 100644
index 0000000..4b10f8a
--- /dev/null
+++ b/simulation/ball.py
@@ -0,0 +1,37 @@
+from geometry.point import Point
+from geometry.vector import Vector
+from geometry.validation import expect_type
+from simulation.drawing import centered_circle
+from turtle import begin_fill, end_fill, fillcolor
+
+class Ball:
+    def __init__(self, position, velocity=None, mass=1, radius=20, color="black"):
+        expect_type(position, Point)
+        self.position = position
+        self.velocity = velocity or Vector()
+        expect_type(self.velocity, Vector)
+        self.acceleration = Vector()
+        self.radius = radius
+        self.mass = mass
+        self.color = color
+
+    def update(self):
+        """Update the ball's position and velocity at this moment. 
+        
+        """
+        self.velocity += self.acceleration
+        self.position += self.velocity
+        self.acceleration = Vector()
+
+    def collided_with(self, other_ball):
+        """Checks whether this ball collided with another ball.
+        """
+        return (self.position - other_ball.position).mag() <= self.radius + other_ball.radius
+
+    def draw(self):
+        "Draws a colored ball."
+        fillcolor(self.color)
+        begin_fill()
+        centered_circle(self.radius)
+        end_fill()
+
diff --git a/simulation/collision.py b/simulation/collision.py
new file mode 100644
index 0000000..7815625
--- /dev/null
+++ b/simulation/collision.py
@@ -0,0 +1,39 @@
+def collide2d(a, b, e):
+    """Returns the velocities of balls a and b after collision. 
+
+    e is the *coefficient of restitution*, a parameter between 0 and 1 
+    which determines how much the objects bounce off each other. 
+    When e is 0, the two objects completely stick together. 
+    When e is 1, the two objects bounce off each other completely. 
+
+    This is a 2-dimensional collision because a and b each have a 2d 
+    velocity vector, like two balls moving on a plane. We can separate
+    each velocity vector into two components: a normal vector (pointed 
+    straight at the other ball) and a tangent vector (perpendicular to
+    the normal). The tangent vectors don't participate in the collision
+    at all, so we just need to calculate how the normal vectors change
+    after the collision. This is easier, since it's a 1-dimensional 
+    collision.
+
+    Afterwards we combine the updated normal vector with the tangent 
+    vector to get the resulting velocity vector for each ball.
+    """
+    a_to_b = b.position - a.position
+    av_normal = a.velocity.project(a_to_b)
+    av_tangent = a.velocity - av_normal
+    bv_normal = b.velocity.project(-a_to_b)
+    bv_tangent = b.velocity - bv_normal
+    av_normal_after, bv_normal_after = collide1d(av_normal, a.mass, bv_normal, b.mass, e)
+    av_after = av_normal_after + av_tangent
+    bv_after = bv_normal_after + bv_tangent
+    return av_after, bv_after
+
+def collide1d(av, am, bv, bm, e):
+    """Returns the velicoties of a and b after a 1-dimensional collision, 
+    where a and b are headed directly at each other. This formula is from Wikipedia: 
+    https://en.wikipedia.org/wiki/Elastic_collision#One-dimensional_Newtonian
+    """
+    velocity_center_of_mass = (av * am + bv * bm) / (am + bm)
+    av_after = velocity_center_of_mass * (1 + e) - av * e
+    bv_after = velocity_center_of_mass * (1 + e) - bv * e
+    return av_after, bv_after
diff --git a/simulation/drawing.py b/simulation/drawing.py
new file mode 100644
index 0000000..226884b
--- /dev/null
+++ b/simulation/drawing.py
@@ -0,0 +1,31 @@
+from turtle import *
+
+def centered_circle(radius):
+    """Draws a circle, centered on the turtle's current position.
+    The built-in circle function annoyingly draws a circle whose center
+    is one radius left of the turtle.
+    """
+    right(90)
+    penup()
+    forward(radius)
+    pendown()
+    left(90)
+    circle(radius)
+    left(90)
+    penup()
+    forward(radius)
+    pendown()
+    right(90)
+
+def draw_boundary(distance):
+    """Draws the boundary, a square `distance` from the origin on each side.
+    """
+    penup()
+    goto(-distance, -distance)
+    pendown()
+    goto(distance, -distance)
+    goto(distance, distance)
+    goto(-distance, distance)
+    goto(-distance, -distance)
+
+
diff --git a/simulation/gravity.py b/simulation/gravity.py
new file mode 100644
index 0000000..9e7cc3f
--- /dev/null
+++ b/simulation/gravity.py
@@ -0,0 +1,13 @@
+
+def get_gravity_force(a, b, g):
+    """Gravity is a force of attraction between objects. 
+    The attraction between two objects is equal to their masses
+    multipled together, divided by the square of the distance between them, 
+    times a constant g. This is the real-life definition of gravity. 
+    """
+    a_to_b = b.position - a.position
+    distance = a_to_b.mag()
+    force = a_to_b.scale(g * a.mass * b.mass / (distance * distance))
+    return force
+    a.acceleration += force
+    b.acceleration -= force
diff --git a/tests/test_point.py b/tests/test_point.py
new file mode 100644
index 0000000..86704ef
--- /dev/null
+++ b/tests/test_point.py
@@ -0,0 +1,48 @@
+from unittest import TestCase
+from geometry.point import Point
+from geometry.vector import Vector
+
+class TestPoint(TestCase):
+    def setUp(self):
+        self.origin = Point((0, 0))
+        self.a = Point((4, 0))
+        self.b = Point((0, 3))
+        self.c = Point((5, 2))
+        self.v = Vector((1, 2))
+
+    def test_init_with_none(self):
+        origin = Point()
+        self.assertEqual(origin.x, 0)
+
+    def test_init_with_pair(self):
+        p = Point((4, 2))
+        self.assertEqual(p.x, 4)
+
+    def test_init_with_vector(self):
+        self.assertEqual(self.a, Point(Vector(self.a)))
+
+    def test_string_representation(self):
+        self.assertEqual(str(self.a), "(4, 0)")
+
+    def test_equality(self):
+        self.assertEqual(self.a, self.a)
+        self.assertNotEqual(self.a, self.b)
+
+    def test_addition_with_vector(self):
+        self.assertEqual(self.a + self.v, self.c)
+
+    def test_addition_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.a + self.a
+
+    def test_subtraction_with_point(self):
+        self.assertEqual(self.c - self.a, self.v)
+
+    def test_subtraction_with_vector(self):
+        self.assertEqual(self.c - self.v, self.a)
+
+    def test_subtraction_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.a - 4
+        
+
diff --git a/tests/test_vector.py b/tests/test_vector.py
new file mode 100644
index 0000000..8e9b757
--- /dev/null
+++ b/tests/test_vector.py
@@ -0,0 +1,101 @@
+from unittest import TestCase
+from geometry.vector import Vector
+from geometry.point import Point
+
+class TestVector(TestCase):
+    def setUp(self):
+        self.zero = Vector()
+        self.unit_x = Vector((1, 0))
+        self.unit_y = Vector((0, 1))
+        self.origin = Point()
+
+    def test_init_with_none(self):
+        self.assertEqual(self.zero.x, 0)
+
+    def test_init_with_pair(self):
+        v = Vector((4, 2))
+        self.assertEqual(v.x, 4)
+
+    def test_init_with_point(self):
+        self.assertEqual(self.unit_x, Vector(Point(self.unit_x)))
+
+    def test_string_representation(self):
+        self.assertEqual(str(self.unit_x), "〈1, 0〉")
+
+    def test_equality(self):
+        self.assertEqual(self.unit_x, self.unit_x)
+        self.assertNotEqual(self.unit_x, self.unit_y)
+
+    def test_negation(self):
+        self.assertEqual(self.zero, -self.zero)
+        self.assertEqual(-self.unit_x, Vector((-1, 0)))
+
+    def test_addition_with_vector(self):
+        self.assertEqual(self.unit_x + self.unit_y, Vector((1, 1)))
+
+    def test_addition_with_point(self):
+        self.assertEqual(self.unit_x + Point((3, 4)), Point((4, 4)))
+
+    def test_addition_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.unit_x + 2
+
+    def test_subtraction_with_vector(self):
+        self.assertEqual(Vector((2, 0)) - self.unit_x, self.unit_x)
+
+    def test_subtraction_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.unit_x - 4
+
+    def test_multiplication_with_numeric(self):
+        self.assertEqual(self.unit_x + self.unit_x, self.unit_x * 2)
+        
+    def test_multiplication_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.unit_x * self.unit_x
+
+    def test_division(self):
+        self.assertEqual(self.unit_x, Vector((2, 0)) / 2)
+
+    def test_division_with_zero(self):
+        with self.assertRaises(ZeroDivisionError):
+            self.unit_x / 0
+
+    def test_division_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.unit_x / self.unit_x
+
+    def test_truthiness(self):
+        self.assertTrue(self.unit_x)
+        self.assertFalse(self.zero)
+
+    def test_magnitude(self):
+        self.assertEqual(self.zero.mag(), 0)
+        self.assertEqual(self.unit_x.mag(), 1)
+        self.assertEqual(Vector((3, 4)).mag(), 5)
+
+    def test_normalize(self):
+        self.assertEqual((self.unit_x * 2).normalize(), self.unit_x)
+
+    def test_normalize_with_zero_vector(self):
+        with self.assertRaises(ValueError):
+            self.zero.normalize()
+
+    def test_is_collinear(self):
+        self.assertTrue(self.unit_x.is_collinear(self.unit_x * -4))
+        self.assertFalse(self.unit_x.is_collinear(self.unit_y))
+
+    def test_is_collinear_with_zero_vector(self):
+        self.assertFalse(self.unit_x.is_collinear(self.zero))
+        self.assertFalse(self.zero.is_collinear(self.zero))
+
+    def test_division_with_invalid_type(self):
+        with self.assertRaises(TypeError):
+            self.unit_x.is_collinear(4)
+
+
+
+
+
+
+