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:
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
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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