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\|}
2D conic section: Solution to Ax^2 + By^2 + Cxy + Dx + Ey + F = 0 (intersection of a plane with double cone)
The first four can be obtained by imagine:
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
Constant Non-Zero (Ellipsoid, all Hyperboloid): homogeneous
Other Types:
Cylinders: y^2 = 4z
Circular Cylinders: y^2 + z^2 = 4
Plane: Non Quadratics Term
Trace: a curve of intersection with a plane (parallel) to a coordinate plane
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