Coordinate Geometry¶
Recall (doc/postdoc). Not a first course.
Job: Cartesian model of Euclidean plane/space; analytic forms of lines, circles, conics.
Core¶
Distance / midpoint in \(\mathbb{R}^n\): \(\|x-y\|_2\); midpoint \((x+y)/2\).
| Object | Equation |
|---|---|
| Line | \(ax+by+c=0\) or parametric \(r=r_0+td\) |
| Slope form | \(y=mx+b\) |
| Circle | \((x-h)^2+(y-k)^2=r^2\) |
| General conic | \(ax^2+bxy+cy^2+dx+ey+f=0\) |
| Discriminant | \(b^2-4ac\): ellipse / parabola / hyperbola (nondeg.) |
Parallel / perp: slopes \(m_1=m_2\); \(m_1m_2=-1\) (Euclidean, nonvertical).
Section formula: point dividing \(AB\) in \(m:n\) is \(\frac{nx_A+mx_B}{m+n}\) (internal).
Used later for¶
- Linear algebra: vectors as coordinates
- Multivariable calculus: graphs, level sets
- Algebraic geometry: affine varieties in \(K^n\)
- Differential geometry: charts \(\mathbb{R}^n\)
Check¶
Does \(b^2-4ac\) classify nondegenerate conics over \(\mathbb{R}\) completely?
Up to projective type for the quadratic form part; full affine classification also needs degeneracy (\(\det\) of \(3\times3\) conic matrix) and whether real points exist (e.g. \(x^2+y^2+1=0\)).