From 27089556acbdcd4f19ef707dde7549b3b962bd82 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Wed, 18 Dec 2024 17:55:25 +0100
Subject: [PATCH 01/13] add parametric function for Chain
---
src/geometries/polytopes.jl | 12 +++++++++++-
1 file changed, 11 insertions(+), 1 deletion(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index ac16288f9..571306cd3 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -8,7 +8,7 @@
We say that a geometry is a K-polytope when it is a collection of "flat" sides
that constitute a `K`-dimensional subspace. They are called chain, polygon and
polyhedron respectively for 1D (`K=1`), 2D (`K=2`) and 3D (`K=3`) subspaces.
-The parameter `K` is also known as the rank or parametric dimension
+The parameter `K` is also known as the rank or parametric dimension
of the polytope (<https://en.wikipedia.org/wiki/Abstract_polytope>).
The term polytope expresses a particular combinatorial structure. A polyhedron,
@@ -156,6 +156,16 @@ function angles(c::Chain)
map(i -> ∠(vs[i - 1], vs[i], vs[i + 1]), i1:i2)
end
+function (c::Chain)(t)
+ if t < 0 || t > 1
+ throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
+ end
+ s = collect(segments(c))
+ N = length(s)
+ k = max(1, ceil(Int, N * t))
+ s[k](N * t - k + 1)
+end
+
# implementations of Chain
include("polytopes/segment.jl")
include("polytopes/rope.jl")
From 9c06204f385975711dfbe14f67815cd0b034952f Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Wed, 18 Dec 2024 18:15:48 +0100
Subject: [PATCH 02/13] add tests
---
test/polytopes.jl | 16 ++++++++++++++++
1 file changed, 16 insertions(+)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 2a50b8baa..137b72187 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -264,6 +264,22 @@ end
r = Ring(merc.([(0, 0), (1, 0), (1, 1), (0, 1)]))
@test crs(centroid(r)) === crs(r)
+ # parameterization
+ ri = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
+ @test ri(T(0)) == latlon(45, 0)
+ @test ri(T(0.25)) == latlon(45, 67.5)
+ @test ri(T(0.5)) == latlon(67.5, 90)
+ @test ri(T(0.75)) == latlon(78.75, 67.5)
+ @test ri(T(1)) == latlon(45, 0)
+
+ ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
+ @test ro(T(0)) == latlon(45, 0)
+ @test ro(T(0.25)) == latlon(45, 45)
+ @test ro(T(0.5)) == latlon(45, 90)
+ @test ro(T(0.75)) == latlon(67.5, 90)
+ @test ro(T(1)) == latlon(90, 90)
+
+
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
ro = Rope(cart.([(1, 1), (2, 2), (3, 3)]))
@test sprint(show, ri) == "Ring((x: 1.0 m, y: 1.0 m), (x: 2.0 m, y: 2.0 m), (x: 3.0 m, y: 3.0 m))"
From 192b40ff830c784c7a11fdeae3e3fc1f20a64dd3 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <51029046+JoshuaLampert@users.noreply.github.com>
Date: Wed, 18 Dec 2024 18:18:16 +0100
Subject: [PATCH 03/13] Apply suggestions from code review
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
---
test/polytopes.jl | 1 -
1 file changed, 1 deletion(-)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 137b72187..9a66e7e1e 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -279,7 +279,6 @@ end
@test ro(T(0.75)) == latlon(67.5, 90)
@test ro(T(1)) == latlon(90, 90)
-
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
ro = Rope(cart.([(1, 1), (2, 2), (3, 3)]))
@test sprint(show, ri) == "Ring((x: 1.0 m, y: 1.0 m), (x: 2.0 m, y: 2.0 m), (x: 3.0 m, y: 3.0 m))"
From 05a195be68055a2da3cf8bcd49a970aac5bc1017 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Wed, 18 Dec 2024 21:53:15 +0100
Subject: [PATCH 04/13] reduce allocations
---
src/geometries/polytopes.jl | 9 +++++----
1 file changed, 5 insertions(+), 4 deletions(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index 571306cd3..7f642b684 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -160,10 +160,11 @@ function (c::Chain)(t)
if t < 0 || t > 1
throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
end
- s = collect(segments(c))
- N = length(s)
- k = max(1, ceil(Int, N * t))
- s[k](N * t - k + 1)
+ v = vertices(c)
+ n = length(v) - !isclosed(c)
+ k = max(1, ceil(Int, n * t))
+ s, _ = iterate(segments(c), k)
+ s(n * t - k + 1)
end
# implementations of Chain
From dd528cc357500bdcd9a6bc428411062e906b0e69 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <51029046+JoshuaLampert@users.noreply.github.com>
Date: Wed, 18 Dec 2024 22:09:14 +0100
Subject: [PATCH 05/13] Update src/geometries/polytopes.jl
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
Co-authored-by: Júlio Hoffimann <julio.hoffimann@gmail.com>
---
src/geometries/polytopes.jl | 3 +--
1 file changed, 1 insertion(+), 2 deletions(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index 7f642b684..80e2b0f98 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -160,8 +160,7 @@ function (c::Chain)(t)
if t < 0 || t > 1
throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
end
- v = vertices(c)
- n = length(v) - !isclosed(c)
+ n = nvertices(c) - !isclosed(c)
k = max(1, ceil(Int, n * t))
s, _ = iterate(segments(c), k)
s(n * t - k + 1)
From fbaaed58e4b8e582f666c916599f2f30ffcd3cdf Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Wed, 18 Dec 2024 22:35:26 +0100
Subject: [PATCH 06/13] fix
---
src/geometries/polytopes.jl | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index 80e2b0f98..c57cd7c17 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -161,9 +161,9 @@ function (c::Chain)(t)
throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
end
n = nvertices(c) - !isclosed(c)
- k = max(1, ceil(Int, n * t))
+ k = min(n - 1, floor(Int, n * t))
s, _ = iterate(segments(c), k)
- s(n * t - k + 1)
+ s(n * t - k)
end
# implementations of Chain
From 28d07dcd6da823ce7504952aeea023f314a357f6 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Thu, 19 Dec 2024 11:59:06 +0100
Subject: [PATCH 07/13] use measure
---
src/geometries/polytopes.jl | 15 +++++++++++----
test/polytopes.jl | 12 ++++++------
2 files changed, 17 insertions(+), 10 deletions(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index c57cd7c17..bd922408e 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -160,10 +160,17 @@ function (c::Chain)(t)
if t < 0 || t > 1
throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
end
- n = nvertices(c) - !isclosed(c)
- k = min(n - 1, floor(Int, n * t))
- s, _ = iterate(segments(c), k)
- s(n * t - k)
+ segs = segments(c)
+ measures = cumsum(measure.(segs))
+ measures /= measures[end]
+ # measures[k] <= t < measures[k + 1]
+ k = searchsortedfirst(measures, t) - 1
+ s, _ = iterate(segs, k)
+ mk = k == 0 ? 0 : measures[k]
+ mk1 = measures[k + 1]
+ # Map t to [0, 1] in the segment
+ t2 = (t - mk) / (mk1 - mk)
+ s(t2)
end
# implementations of Chain
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 9a66e7e1e..7fe876820 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -267,16 +267,16 @@ end
# parameterization
ri = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ri(T(0)) == latlon(45, 0)
- @test ri(T(0.25)) == latlon(45, 67.5)
- @test ri(T(0.5)) == latlon(67.5, 90)
- @test ri(T(0.75)) == latlon(78.75, 67.5)
+ @test ri(T(0.25)) ≈ latlon(45, 56.25) atol=1e7 * eps(T) * u"m"
+ @test ri(T(0.5)) ≈ latlon(60, 90) atol=1e7 * eps(T) * u"m"
+ @test ri(T(0.75)) ≈ latlon(82.5, 75) atol=1e7 * eps(T) * u"m"
@test ri(T(1)) == latlon(45, 0)
ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ro(T(0)) == latlon(45, 0)
- @test ro(T(0.25)) == latlon(45, 45)
- @test ro(T(0.5)) == latlon(45, 90)
- @test ro(T(0.75)) == latlon(67.5, 90)
+ @test ro(T(0.25)) ≈ latlon(45, 39.375) atol=1e7 * eps(T) * u"m"
+ @test ro(T(0.5)) ≈ latlon(45, 78.75) atol=1e7 * eps(T) * u"m"
+ @test ro(T(0.75)) ≈ latlon(63.75, 90) atol=1e7 * eps(T) * u"m"
@test ro(T(1)) == latlon(90, 90)
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
From a45df4fc57e569bb04bc81fb13a868735fbd35d8 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <51029046+JoshuaLampert@users.noreply.github.com>
Date: Thu, 19 Dec 2024 12:00:03 +0100
Subject: [PATCH 08/13] Apply suggestions from code review
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
---
test/polytopes.jl | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 7fe876820..7810657d3 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -267,16 +267,16 @@ end
# parameterization
ri = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ri(T(0)) == latlon(45, 0)
- @test ri(T(0.25)) ≈ latlon(45, 56.25) atol=1e7 * eps(T) * u"m"
- @test ri(T(0.5)) ≈ latlon(60, 90) atol=1e7 * eps(T) * u"m"
- @test ri(T(0.75)) ≈ latlon(82.5, 75) atol=1e7 * eps(T) * u"m"
+ @test ri(T(0.25)) ≈ latlon(45, 56.25) atol = 1e7 * eps(T) * u"m"
+ @test ri(T(0.5)) ≈ latlon(60, 90) atol = 1e7 * eps(T) * u"m"
+ @test ri(T(0.75)) ≈ latlon(82.5, 75) atol = 1e7 * eps(T) * u"m"
@test ri(T(1)) == latlon(45, 0)
ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ro(T(0)) == latlon(45, 0)
- @test ro(T(0.25)) ≈ latlon(45, 39.375) atol=1e7 * eps(T) * u"m"
- @test ro(T(0.5)) ≈ latlon(45, 78.75) atol=1e7 * eps(T) * u"m"
- @test ro(T(0.75)) ≈ latlon(63.75, 90) atol=1e7 * eps(T) * u"m"
+ @test ro(T(0.25)) ≈ latlon(45, 39.375) atol = 1e7 * eps(T) * u"m"
+ @test ro(T(0.5)) ≈ latlon(45, 78.75) atol = 1e7 * eps(T) * u"m"
+ @test ro(T(0.75)) ≈ latlon(63.75, 90) atol = 1e7 * eps(T) * u"m"
@test ro(T(1)) == latlon(90, 90)
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
From ae84e58a8400a6befb8744a14730a324c4cd08b1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?J=C3=BAlio=20Hoffimann?= <julio.hoffimann@gmail.com>
Date: Thu, 19 Dec 2024 09:29:29 -0300
Subject: [PATCH 09/13] Fix type instabilities
---
src/geometries/polytopes.jl | 18 +++++++++---------
1 file changed, 9 insertions(+), 9 deletions(-)
diff --git a/src/geometries/polytopes.jl b/src/geometries/polytopes.jl
index bd922408e..463011bbc 100644
--- a/src/geometries/polytopes.jl
+++ b/src/geometries/polytopes.jl
@@ -161,16 +161,16 @@ function (c::Chain)(t)
throw(DomainError(t, "c(t) is not defined for t outside [0, 1]."))
end
segs = segments(c)
- measures = cumsum(measure.(segs))
- measures /= measures[end]
- # measures[k] <= t < measures[k + 1]
- k = searchsortedfirst(measures, t) - 1
+ sums = cumsum(measure.(segs))
+ sums /= last(sums)
+ # find k such that sums[k] ≤ t < sums[k + 1]
+ k = searchsortedfirst(sums, t) - 1
+ # select segment s at index k
s, _ = iterate(segs, k)
- mk = k == 0 ? 0 : measures[k]
- mk1 = measures[k + 1]
- # Map t to [0, 1] in the segment
- t2 = (t - mk) / (mk1 - mk)
- s(t2)
+ # reparametrization of t within s
+ ∑ₖ = iszero(k) ? zero(eltype(sums)) : sums[k]
+ ∑ₖ₊₁ = sums[k + 1]
+ s((t - ∑ₖ) / (∑ₖ₊₁ - ∑ₖ))
end
# implementations of Chain
From 44c227040cb63b1b0c33df0fc9ff57dc8c434d81 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Thu, 19 Dec 2024 13:44:00 +0100
Subject: [PATCH 10/13] add tests for Cartesian coords
---
test/polytopes.jl | 13 +++++++++++++
1 file changed, 13 insertions(+)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 7810657d3..38e63f5c0 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -272,6 +272,12 @@ end
@test ri(T(0.75)) ≈ latlon(82.5, 75) atol = 1e7 * eps(T) * u"m"
@test ri(T(1)) == latlon(45, 0)
+ ri = boundary(Box(cart(0,0), cart(1,1)))
+ @test ri(T(0)) == cart(0, 0)
+ @test ri(T(0.25)) == cart(1, 0)
+ @test ri(T(0.5)) == cart(1, 1)
+ @test ri(T(0.75)) == cart(0, 1)
+
ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ro(T(0)) == latlon(45, 0)
@test ro(T(0.25)) ≈ latlon(45, 39.375) atol = 1e7 * eps(T) * u"m"
@@ -279,6 +285,13 @@ end
@test ro(T(0.75)) ≈ latlon(63.75, 90) atol = 1e7 * eps(T) * u"m"
@test ro(T(1)) == latlon(90, 90)
+ ro = Rope(cart(0,0), cart(3, 0), cart(4,0))
+ @test ro(T(0)) == cart(0, 0)
+ @test ro(T(0.25)) == cart(1, 0)
+ @test ro(T(0.5)) == cart(2, 0)
+ @test ro(T(0.75)) == cart(3, 0)
+ @test ro(T(1)) == cart(4, 0)
+
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
ro = Rope(cart.([(1, 1), (2, 2), (3, 3)]))
@test sprint(show, ri) == "Ring((x: 1.0 m, y: 1.0 m), (x: 2.0 m, y: 2.0 m), (x: 3.0 m, y: 3.0 m))"
From 08c112aa7fa78b8ec38539b8c8bf962d1cb8cda4 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <51029046+JoshuaLampert@users.noreply.github.com>
Date: Thu, 19 Dec 2024 13:45:18 +0100
Subject: [PATCH 11/13] Apply suggestions from code review
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
---
test/polytopes.jl | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 38e63f5c0..cdc403944 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -272,7 +272,7 @@ end
@test ri(T(0.75)) ≈ latlon(82.5, 75) atol = 1e7 * eps(T) * u"m"
@test ri(T(1)) == latlon(45, 0)
- ri = boundary(Box(cart(0,0), cart(1,1)))
+ ri = boundary(Box(cart(0, 0), cart(1, 1)))
@test ri(T(0)) == cart(0, 0)
@test ri(T(0.25)) == cart(1, 0)
@test ri(T(0.5)) == cart(1, 1)
@@ -285,7 +285,7 @@ end
@test ro(T(0.75)) ≈ latlon(63.75, 90) atol = 1e7 * eps(T) * u"m"
@test ro(T(1)) == latlon(90, 90)
- ro = Rope(cart(0,0), cart(3, 0), cart(4,0))
+ ro = Rope(cart(0, 0), cart(3, 0), cart(4, 0))
@test ro(T(0)) == cart(0, 0)
@test ro(T(0.25)) == cart(1, 0)
@test ro(T(0.5)) == cart(2, 0)
From bcaf0f261475c278e91e8602d77ee761b4386766 Mon Sep 17 00:00:00 2001
From: Joshua Lampert <joshua.lampert@uni-hamburg.de>
Date: Thu, 19 Dec 2024 13:46:20 +0100
Subject: [PATCH 12/13] remove approximate latlon tests
---
test/polytopes.jl | 6 ------
1 file changed, 6 deletions(-)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 38e63f5c0..cfe71b205 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -267,9 +267,6 @@ end
# parameterization
ri = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ri(T(0)) == latlon(45, 0)
- @test ri(T(0.25)) ≈ latlon(45, 56.25) atol = 1e7 * eps(T) * u"m"
- @test ri(T(0.5)) ≈ latlon(60, 90) atol = 1e7 * eps(T) * u"m"
- @test ri(T(0.75)) ≈ latlon(82.5, 75) atol = 1e7 * eps(T) * u"m"
@test ri(T(1)) == latlon(45, 0)
ri = boundary(Box(cart(0,0), cart(1,1)))
@@ -280,9 +277,6 @@ end
ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
@test ro(T(0)) == latlon(45, 0)
- @test ro(T(0.25)) ≈ latlon(45, 39.375) atol = 1e7 * eps(T) * u"m"
- @test ro(T(0.5)) ≈ latlon(45, 78.75) atol = 1e7 * eps(T) * u"m"
- @test ro(T(0.75)) ≈ latlon(63.75, 90) atol = 1e7 * eps(T) * u"m"
@test ro(T(1)) == latlon(90, 90)
ro = Rope(cart(0,0), cart(3, 0), cart(4,0))
From e4011bdb787d45e8ea8f8908aaa1075d81ba0179 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?J=C3=BAlio=20Hoffimann?= <julio.hoffimann@gmail.com>
Date: Thu, 19 Dec 2024 09:50:05 -0300
Subject: [PATCH 13/13] Adjust tests
---
test/polytopes.jl | 37 +++++++++++++++++--------------------
1 file changed, 17 insertions(+), 20 deletions(-)
diff --git a/test/polytopes.jl b/test/polytopes.jl
index 890fe6717..d677e6433 100644
--- a/test/polytopes.jl
+++ b/test/polytopes.jl
@@ -265,26 +265,23 @@ end
@test crs(centroid(r)) === crs(r)
# parameterization
- ri = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
- @test ri(T(0)) == latlon(45, 0)
- @test ri(T(1)) == latlon(45, 0)
-
- ri = boundary(Box(cart(0, 0), cart(1, 1)))
- @test ri(T(0)) == cart(0, 0)
- @test ri(T(0.25)) == cart(1, 0)
- @test ri(T(0.5)) == cart(1, 1)
- @test ri(T(0.75)) == cart(0, 1)
-
- ro = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
- @test ro(T(0)) == latlon(45, 0)
- @test ro(T(1)) == latlon(90, 90)
-
- ro = Rope(cart(0, 0), cart(3, 0), cart(4, 0))
- @test ro(T(0)) == cart(0, 0)
- @test ro(T(0.25)) == cart(1, 0)
- @test ro(T(0.5)) == cart(2, 0)
- @test ro(T(0.75)) == cart(3, 0)
- @test ro(T(1)) == cart(4, 0)
+ r = boundary(Box(cart(0, 0), cart(1, 1)))
+ @test r(T(0)) == cart(0, 0)
+ @test r(T(0.25)) == cart(1, 0)
+ @test r(T(0.5)) == cart(1, 1)
+ @test r(T(0.75)) == cart(0, 1)
+ r = Rope(cart(0, 0), cart(3, 0), cart(4, 0))
+ @test r(T(0)) == cart(0, 0)
+ @test r(T(0.25)) == cart(1, 0)
+ @test r(T(0.5)) == cart(2, 0)
+ @test r(T(0.75)) == cart(3, 0)
+ @test r(T(1)) == cart(4, 0)
+ r = Ring(latlon(45, 0), latlon(45, 90), latlon(90, 90))
+ @test r(T(0)) == latlon(45, 0)
+ @test r(T(1)) == latlon(45, 0)
+ r = Rope(latlon(45, 0), latlon(45, 90), latlon(90, 90))
+ @test r(T(0)) == latlon(45, 0)
+ @test r(T(1)) == latlon(90, 90)
ri = Ring(cart.([(1, 1), (2, 2), (3, 3)]))
ro = Rope(cart.([(1, 1), (2, 2), (3, 3)]))