How can Newell's method for determining a plane equation be used to check for abnormal inputs?
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Newell's method for obtaining a plane equation $$ax + bx + cz +d = 0$$ for $n$ points has the factors determined as
$$a = sum_{i=0}^{n}(y_{i} - y_{i+1})(z_{i} + z_{i+1})$$
$$b = sum_{i=0}^{n}(z_{i} - z_{i+1})(x_{i} + x_{i+1})$$
$$c = sum_{i=0}^{n}(x_{i} - x_{i+1})(y_{i} + y_{i+1})$$
$$d = - frac{1}{n} sum_{i=0}^{n} V_i cdot [a, b, c]^T $$
where point index addition is performed modulo $n$.
See the full description here.
How can this method be used (without further extensive calculations) to check for the following abnormal inputs:
- all points are colinear,
- all points are identical,
- some points do not lie on the same plane. (I realize that the method is written to work even if this is the case, but it would be nice to be able to have a metric identifying the points' fit.)
linear-algebra algebraic-geometry
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
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add a comment |
$begingroup$
Newell's method for obtaining a plane equation $$ax + bx + cz +d = 0$$ for $n$ points has the factors determined as
$$a = sum_{i=0}^{n}(y_{i} - y_{i+1})(z_{i} + z_{i+1})$$
$$b = sum_{i=0}^{n}(z_{i} - z_{i+1})(x_{i} + x_{i+1})$$
$$c = sum_{i=0}^{n}(x_{i} - x_{i+1})(y_{i} + y_{i+1})$$
$$d = - frac{1}{n} sum_{i=0}^{n} V_i cdot [a, b, c]^T $$
where point index addition is performed modulo $n$.
See the full description here.
How can this method be used (without further extensive calculations) to check for the following abnormal inputs:
- all points are colinear,
- all points are identical,
- some points do not lie on the same plane. (I realize that the method is written to work even if this is the case, but it would be nice to be able to have a metric identifying the points' fit.)
linear-algebra algebraic-geometry
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
$endgroup$
add a comment |
$begingroup$
Newell's method for obtaining a plane equation $$ax + bx + cz +d = 0$$ for $n$ points has the factors determined as
$$a = sum_{i=0}^{n}(y_{i} - y_{i+1})(z_{i} + z_{i+1})$$
$$b = sum_{i=0}^{n}(z_{i} - z_{i+1})(x_{i} + x_{i+1})$$
$$c = sum_{i=0}^{n}(x_{i} - x_{i+1})(y_{i} + y_{i+1})$$
$$d = - frac{1}{n} sum_{i=0}^{n} V_i cdot [a, b, c]^T $$
where point index addition is performed modulo $n$.
See the full description here.
How can this method be used (without further extensive calculations) to check for the following abnormal inputs:
- all points are colinear,
- all points are identical,
- some points do not lie on the same plane. (I realize that the method is written to work even if this is the case, but it would be nice to be able to have a metric identifying the points' fit.)
linear-algebra algebraic-geometry
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
$endgroup$
Newell's method for obtaining a plane equation $$ax + bx + cz +d = 0$$ for $n$ points has the factors determined as
$$a = sum_{i=0}^{n}(y_{i} - y_{i+1})(z_{i} + z_{i+1})$$
$$b = sum_{i=0}^{n}(z_{i} - z_{i+1})(x_{i} + x_{i+1})$$
$$c = sum_{i=0}^{n}(x_{i} - x_{i+1})(y_{i} + y_{i+1})$$
$$d = - frac{1}{n} sum_{i=0}^{n} V_i cdot [a, b, c]^T $$
where point index addition is performed modulo $n$.
See the full description here.
How can this method be used (without further extensive calculations) to check for the following abnormal inputs:
- all points are colinear,
- all points are identical,
- some points do not lie on the same plane. (I realize that the method is written to work even if this is the case, but it would be nice to be able to have a metric identifying the points' fit.)
linear-algebra algebraic-geometry
linear-algebra algebraic-geometry
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
asked Dec 15 '18 at 16:34
Diomidis SpinellisDiomidis Spinellis
1012
1012
add a comment |
add a comment |
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