tadd conformal mapping functions - Granular.jl - Julia package for granular dynamics simulation
(HTM) git clone git://src.adamsgaard.dk/Granular.jl
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---
(DIR) commit 94e04429fe360c2a5fec893229973145516381de
(DIR) parent abeaa9bfc5a527df9a1bb1fbf63aa580a0139971
(HTM) Author: Anders Damsgaard <andersd@riseup.net>
Date: Wed, 26 Apr 2017 15:59:58 -0400
add conformal mapping functions
Diffstat:
M src/grid.jl | 106 ++++++++++++++++++++++++++-----
M test/grid.jl | 31 +++++++++++++++++++++++++++++++
2 files changed, 122 insertions(+), 15 deletions(-)
---
(DIR) diff --git a/src/grid.jl b/src/grid.jl
t@@ -65,34 +65,50 @@ end
"""
-Check if a 2d point is contained inside a cell from the ocean grid. Returns
-`true`/`false`.
+Check if a 2d point is contained inside a cell from the ocean grid.
+The function uses either an area-based approach (`method = "Area"`), or a
+conformal mapping approach (`method = "Conformal"`). The area-based approach is
+more robust. This function returns `true` or `false`.
"""
-function isPointInCell(ocean::Ocean, i::Int, j::Int, point::Array{float, 1})
-
- sw, nw, se, ne = getCellCornerCoordinates(ocean, i, j)
+function isPointInCell(ocean::Ocean, i::Int, j::Int, point::Array{float, 1};
+ method::String="Area")
+
+ sw, se, ne, nw = getCellCornerCoordinates(ocean, i, j)
+
+ if method == "Area"
+ if areaOfQuadrilateral(sw, se, ne, nw) ≈
+ areaOfTriangle(point, sw, se) +
+ areaOfTriangle(point, se, ne) +
+ areaOfTriangle(point, ne, nw) +
+ areaOfTriangle(point, nw, sw)
+ return true
+ else
+ return false
+ end
- if areaOfQuadrilateral(sw, nw, se, ne) ≈
- areaOfTriangle(point, sw, se) +
- areaOfTriangle(point, se, ne) +
- areaOfTriangle(point, ne, nw) +
- areaOfTriangle(point, nw, sw)
- return true
+ elseif method == "Conformal"
+ x_tilde, y_tilde = conformalQuadrilateralCoordinates(sw, se, ne, nw,
+ point)
+ if x_tilde >= 0. && x_tilde <= 1. && y_tilde >= 0. && y_tilde <= 1.
+ return true
+ else
+ return false
+ end
else
- return false
+ error("method not understood")
end
end
"""
Returns ocean-grid corner coordinates in the following order (south-west corner,
-north-west corner, south-east corner, north-east corner).
+south-east corner, north-east corner, north-west corner).
"""
function getCellCornerCoordinates(ocean::Ocean, i::Int, j::Int)
sw = [ocean.xq[i-1, j-1], ocean.yq[i-1, j-1]]
- nw = [ocean.xq[i-1, j], ocean.yq[i-1, j]]
se = [ocean.xq[ i, j-1], ocean.yq[ i, j-1]]
ne = [ocean.xq[ i, j], ocean.yq[ i, j]]
- return sw, nw, se, ne
+ nw = [ocean.xq[i-1, j], ocean.yq[i-1, j]]
+ return sw, se, ne, nw
end
"Returns the area of an triangle with corner coordinates `a`, `b`, and `c`."
t@@ -119,3 +135,63 @@ function areaOfQuadrilateral(a::Array{float, 1},
return areaOfTriangle(a, b, c) + areaOfTriangle(c, d, a)
end
+"""
+Returns the non-dimensional coordinates `[x_tilde, y_tilde]` of a point `p`
+within a quadrilateral with corner coordinates `A`, `B`, `C`, and `D`.
+Points must be ordered in counter-clockwise order, starting from south-west
+corner.
+"""
+function conformalQuadrilateralCoordinates(A::Array{float, 1},
+ B::Array{float, 1},
+ C::Array{float, 1},
+ D::Array{float, 1},
+ p::Array{float, 1})
+ alpha = B[1] - A[1]
+ delta = B[2] - A[2]
+ beta = D[1] - A[1]
+ epsilon = D[2] - A[2]
+ gamma = C[1] - A[1] - (alpha + beta)
+ kappa = C[2] - A[2] - (delta + epsilon)
+ a = kappa*beta - gamma*epsilon
+ dx = p[1] - A[1]
+ dy = p[2] - A[2]
+ b = (delta*beta - alpha*epsilon) - (kappa*dx - gamma*dy)
+ c = alpha*dy - delta*dx
+ if abs(a) > 0.
+ d = b^2./4. - a*c
+ if d >= 0.
+ yy1 = -(b/2. + sqrt(d))/a
+ yy2 = -(b/2. - sqrt(d))/a
+ if abs(yy1 - .5) < abs(yy2 - .5)
+ y_tilde = yy1
+ else
+ y_tilde = yy2
+ end
+ else
+ error("could not perform conformal mapping\n",
+ "A = $(A), B = $(B), C = $(C), D = $(D), point = $(p),\n",
+ "alpha = $(alpha), beta = $(beta), gamma = $(gamma), ",
+ "delta = $(delta), epsilon = $(epsilon), kappa = $(kappa)")
+ end
+ else
+ if !(b ≈ 0.)
+ y_tilde = -c/b
+ else
+ y_tilde = 0.
+ end
+ end
+ a = alpha + gamma*y_tilde
+ b = delta + kappa*y_tilde
+ if !(a ≈ 0.)
+ x_tilde = (dx - beta*y_tilde)/a
+ elseif !(b ≈ 0.)
+ x_tilde = (dy - epsilon*y_tilde)/b
+ else
+ error("could not determine non-dimensional position in quadrilateral ",
+ "(a = 0. and b = 0.)\n",
+ "A = $(A), B = $(B), C = $(C), D = $(D), point = $(p),\n",
+ "alpha = $(alpha), beta = $(beta), gamma = $(gamma), ",
+ "delta = $(delta), epsilon = $(epsilon), kappa = $(kappa)")
+ end
+ return [x_tilde, y_tilde]
+end
(DIR) diff --git a/test/grid.jl b/test/grid.jl
t@@ -7,9 +7,12 @@ info("#### $(basename(@__FILE__)) ####")
ocean = SeaIce.readOceanNetCDF("Baltic/00010101.ocean_month.nc",
"Baltic/ocean_hgrid.nc")
+info("Testing area-determination methods")
@test SeaIce.areaOfTriangle([0., 0.], [1., 0.], [0., 1.]) ≈ .5
@test SeaIce.areaOfTriangle([1., 0.], [0., 1.], [0., 0.]) ≈ .5
@test SeaIce.areaOfQuadrilateral([1., 0.], [0., 1.], [0., 0.], [1., 1.]) ≈ 1.
+
+info("Testing area-based cell content determination")
@test SeaIce.isPointInCell(ocean, 2, 2, [6.5, 53.5]) == true
@test SeaIce.isPointInCell(ocean, 2, 2, [6.1, 53.5]) == true
@test SeaIce.isPointInCell(ocean, 2, 2, [6.0, 53.5]) == true
t@@ -19,3 +22,31 @@ ocean = SeaIce.readOceanNetCDF("Baltic/00010101.ocean_month.nc",
@test SeaIce.isPointInCell(ocean, 2, 2, [7.5, 53.5]) == false
@test SeaIce.isPointInCell(ocean, 2, 2, [0.0, 53.5]) == false
+info("Testing conformal mapping methods")
+@test SeaIce.conformalQuadrilateralCoordinates([0., 0.],
+ [5., 0.],
+ [5., 3.],
+ [0., 3.],
+ [2.5, 1.5]) ≈ [0.5, 0.5]
+@test SeaIce.conformalQuadrilateralCoordinates([0., 0.],
+ [5., 0.],
+ [5., 3.],
+ [0., 3.],
+ [7.5, 1.5]) ≈ [1.5, 0.5]
+@test SeaIce.conformalQuadrilateralCoordinates([0., 0.],
+ [5., 0.],
+ [5., 3.],
+ [0., 3.],
+ [7.5,-1.5]) ≈ [1.5,-0.5]
+info("Checking cell content using conformal mapping methods")
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.4, 53.4], method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.1, 53.5], method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.0, 53.5], method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.1, 53.7], method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.1, 53.9], method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [6.1, 53.99999],
+ method="Conformal") == true
+@test SeaIce.isPointInCell(ocean, 2, 2, [7.5, 53.5],
+ method="Conformal") == false
+@test SeaIce.isPointInCell(ocean, 2, 2, [0.0, 53.5],
+ method="Conformal") == false