Lecture 006

Distance

Find the distance between a point $Q$ and a line $L$: - Let line $L = \overrightarrow{Q} + t \overrightarrow{v}$ - then the distance $\frac{\|\overrightarrow{QP} \times \overrightarrow{v}\|}{\|\overrightarrow{v}\|}$ - this can be proved by constructing parallelogram with $\overrightarrow{QP}$ and $\overrightarrow{v}$ where $\overrightarrow{v}$ is starts from $Q$ by definition. - think about projection for vector $Q$ on line $L$. However, instead of taking the projection (dot product), we take cross product due to taking opposite.

Distance between a plane ($\overrightarrow{n}$ with point $Q$) and a point ($P$) - $d = \|\text{proj}_n{\overrightarrow{QP}}\| = |\text{comp}_n \overrightarrow{QP}| = \frac{|\overrightarrow{QP} \cdot n|}{\|n\|}$

Conic Section

2D

2D conic section: Solution to $Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$ (intersection of a plane with double cone)

3D

The first four can be obtained by imagine:

1. The coefficient of a variable change from positive to negative indicate breaking a "wall" on that axis. If no "walls" are broken, that's a ellipse.
2. Breaking 1 "wall" suggest that the axis never hits the surface, forms hyperboloid of one sheet
3. Breaking 2 "wall"s on $x$ and $y$ direction forms "hyperboloid of two sheets"
4. Now, the constant term indicates how "separate" two surfaces are. When it is positive, two surface separate (if all other are positive in normal case). When it is zero, it becomes sort of like "elliptic cone". When it is negative, it inverse other terms, so we can rearrange to make it positive.

Other than the elliptic cone (which is taken to be nonhomogeneous for more subtle reasons) the others are homogeneous if all three variables appear to the second power and nonhomogeneous if at least one of them does not appear to the second power.

3D conic section: $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Jz + K = 0$

• Ellipsoid (point)

• Hyperboloid of 2 sheets

• Hyperboloid of 1 sheet

• Elliptic Cone

• Elliptic Paraboloid

• Hyperbolic Paraboloid

Homogeneous (definition in polynomial): all variable to highest power, all constant non zero

Classify 3D Conics:

• Constant Zero (Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid): non-homogeneous

  • All Variables 2 Power: Elliptic Cone
  • One of Variable Only 1 Power
    • Difference Relation between Other Variables: Elliptic Cone
    • Add Relation between Other Variables: Elliptic Paraboloid

• Constant Non-Zero (Ellipsoid, all Hyperboloid): homogeneous

  • Has No Minus: Ellipsoid
  • Has One Minus: Hyperboloid with One Sheet
  • Has Two Minus: Hyperboloid with Two Sheets
  • Has Three Minus: No Solution

Other Types:

• Cylinders: $y^2 = 4z$

• Circular Cylinders: $y^2 + z^2 = 4$

• Plane: Non Quadratics Term

Traces

Trace: a curve of intersection with a plane (parallel) to a coordinate plane

• trace reduce the dimension of original object