MATH 251 Lecture 5
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Curves and Surfaces
- curves
- 2 equations in (x,y,z)
- 1 independent variable (parameterized)
- surfaces
- 1 equation in (x,y,z)
- 2 independent variables (2 parameters)
Parameterizing a curve/line
- a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and a direction vector
- two points (convert to point and direction vector with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \overrightarrow{Q-P})}
Given a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and a direction vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} , the parametric form of a line is
In general, a parameterized curve consists of 3 functions that give a position in space.
Tangent Line
To get a vector tangent to a curve in 3D space, take the derivative of each component in the parametric form.
Here's an elliptical helix:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r} \left( t \right) = \left( 2\cos{t},\, \sin{t},\, t \right)}
The tangent vector will represent the velocity at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} \left(t\right) = \vec{r}' \left( t \right) = \left( -2\sin{t},\, \cos{t},\, 1 \right)}
The second derivative will be the acceleration:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a} \left( t \right) = \vec{v}' \left( t \right) = \vec{r} '' \left( t \right) = \left( -2\cos{t},\, -\sin{t},\, 0 \right)}
Planes
- Three noncolinear points determine a plane.
- Point and a normal vector
Given:
| Arbitrary point on plane: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{X} = \left\langle x, y, z \right\rangle} |
|---|---|
| Point (on plane): | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{P} = \left\langle x_0, y_0, z_0 \right\rangle} |
| Normal Vector: | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n} = \left\langle p, q, r \right\rangle} |
- The vector form of a plane is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\vec{X} - \vec{P} \right) \cdot \vec{n} = 0}
- the normal form equation of a plane is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \left( x-x_0 \right) + q \left( y-y_0 \right) + r \left( z - z_0 \right) = 0}
Distance between point and Plane
Find the distance between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = (1, -1, 3)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x+2y-z=2} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n} = \left\langle 4,\, 2,\, -1 \right\rangle}
Given any point on the plane—Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = (0, 0, -2)} , for example—the distance is given by: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{comp}_\vec{n} \left(P-Q \right) = \left(P-Q\right) \cdot \hat{n}} .