Coordinate Geometry¶
The Cartesian Coordinate System¶
The plane \(\mathbb{R}^2\) is described by two perpendicular axes (\(x\)-axis, \(y\)-axis) with common origin \(O = (0, 0)\). Every point has a unique coordinate pair \((x, y)\).
Distance and Slope¶
Distance Between Two Points¶
For \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\):
This follows directly from the Pythagorean theorem.
Example. \(d((1, 2), (4, 6)) = \sqrt{9 + 16} = \sqrt{25} = 5\).
Slope of a Line¶
The slope \(m\) of a line through \(P_1\) and \(P_2\) (with \(x_1 \neq x_2\)):
Lines¶
Line Equations¶
Common forms of a line equation:
- Slope-intercept form: \(y = mx + b\) (slope \(m\), \(y\)-intercept \(b\))
- Point-slope form: \(y - y_1 = m(x - x_1)\)
Example. Line through \((1, 3)\) with slope \(2\): \(y - 3 = 2(x - 1)\), so \(y = 2x + 1\).
Parallel and Perpendicular Lines¶
- Parallel: \(m_1 = m_2\)
- Perpendicular: \(m_1 \cdot m_2 = -1\)
Curves in the Plane¶
A curve is the set of all points \((x, y)\) satisfying an equation \(F(x, y) = 0\).
Circle¶
Center \((a, b)\), radius \(r\).
Parabola¶
Example. \(y = x^2\) — parabola with vertex at the origin, opening upward.
Elliptic Curves (Preview)¶
Equations of the form \(y^2 = x^3 + ax + b\) define elliptic curves. These play a central role in modern number theory and in the proof of Fermat's Last Theorem.
Summary¶
| Object | Equation |
|---|---|
| Distance | \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) |
| Line | \(y = mx + b\) |
| Circle | \((x-a)^2 + (y-b)^2 = r^2\) |
| Parabola | \(y = ax^2 + bx + c\) |
| Elliptic curve | \(y^2 = x^3 + ax + b\) |
References¶
- Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996. Chapter 4.