Introduction to signed distance function:
The signed distance function is of a set S in the metric space that determines how the given the point x is close to the boundary of set S with the point x is inside S means the function having the positive values, it decreases in value as x approaches the boundary of S where the signed distance function is zero, and the point x having negative values outside of S. I like to share this Rules of Differentiation with you all through my article.
Formula for Signed Distance Function:
If the point (X, d) is a metric space means the signed distance function f is ,
`g(x)={b(x,s^c) if x in S,-b(x,S) if x in S^(c):}`
where
`b(x,S)=("inf"_(y in S))( b(x,y))`
here 'inf' represents the infimum or the greatest lower bound of a subset S.If S is denoted a subset of the Euclidean space Rn with piecewise smooth boundary, the signed distance function is differentiable, and its gradient(`grad`)satisfies the non linear partial differential equation that is eikonal equation.
`|grad g|=1`
The form of the eikonal equation is
`|grad u(x)|= G(x) ,x in Omega`
It is subject to `u|_(del Omega)=0` , where Ω is an open set in with well behaved boundary, G(x) is a function and having the positive values, `grad` is the gradient. In some case when Fg= 1, the solution gives the signed distance function from `del Omega.`
Example 1 for Signed Distance Function :
Solve `G(x)={x^2y ,if x =1 ; y=1` using signed distance function.
Solution:
`|grad g|=1`
The formula is
`grad g= (del g)/(del x) +(del g)/(del y)`
`(del g)/(del x)=2xy` ; `(del g)/(del y)=x^2(1)=x^2`
subsitute x=1 and y=1 and we get,
`grad g= (del g)/(del x) +(del g)/(del y)`
=2(1)(!)+(1)^2
=2+1=3
The result is `|grad g|=3`
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Example 2 for Signed Distance Function :
Solve `G(x)={x^2 +y^2 if x=1,y=1 ` using signed distance function.
Solution:
`|grad g|=1`
The formula is
`grad g= (del g)/(del x) +(del g)/(del y)`
`(del g)/(del x)=2x` ; `(del g)/(del y)=2y`
subsitute x=1 and y=1 and we get,
`grad g= (del g)/(del x) +(del g)/(del y)`
=2(1)+2(1)
=2+2=4
The result is `|grad g|=4`
The signed distance function is of a set S in the metric space that determines how the given the point x is close to the boundary of set S with the point x is inside S means the function having the positive values, it decreases in value as x approaches the boundary of S where the signed distance function is zero, and the point x having negative values outside of S. I like to share this Rules of Differentiation with you all through my article.
Formula for Signed Distance Function:
If the point (X, d) is a metric space means the signed distance function f is ,
`g(x)={b(x,s^c) if x in S,-b(x,S) if x in S^(c):}`
where
`b(x,S)=("inf"_(y in S))( b(x,y))`
here 'inf' represents the infimum or the greatest lower bound of a subset S.If S is denoted a subset of the Euclidean space Rn with piecewise smooth boundary, the signed distance function is differentiable, and its gradient(`grad`)satisfies the non linear partial differential equation that is eikonal equation.
`|grad g|=1`
The form of the eikonal equation is
`|grad u(x)|= G(x) ,x in Omega`
It is subject to `u|_(del Omega)=0` , where Ω is an open set in with well behaved boundary, G(x) is a function and having the positive values, `grad` is the gradient. In some case when Fg= 1, the solution gives the signed distance function from `del Omega.`
Example 1 for Signed Distance Function :
Solve `G(x)={x^2y ,if x =1 ; y=1` using signed distance function.
Solution:
`|grad g|=1`
The formula is
`grad g= (del g)/(del x) +(del g)/(del y)`
`(del g)/(del x)=2xy` ; `(del g)/(del y)=x^2(1)=x^2`
subsitute x=1 and y=1 and we get,
`grad g= (del g)/(del x) +(del g)/(del y)`
=2(1)(!)+(1)^2
=2+1=3
The result is `|grad g|=3`
Please express your views of this topic what shape has 5 sides by commenting on blog.
Example 2 for Signed Distance Function :
Solve `G(x)={x^2 +y^2 if x=1,y=1 ` using signed distance function.
Solution:
`|grad g|=1`
The formula is
`grad g= (del g)/(del x) +(del g)/(del y)`
`(del g)/(del x)=2x` ; `(del g)/(del y)=2y`
subsitute x=1 and y=1 and we get,
`grad g= (del g)/(del x) +(del g)/(del y)`
=2(1)+2(1)
=2+2=4
The result is `|grad g|=4`
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