Strange stopwatch

config.json

@@ -189,6 +189,18 @@
         ],
         "status": "wip"
       },
+      {
+        "slug": "strange-stopwatch",
+        "name": "Strange Stopwatch",
+        "uuid": "6428590f-2bf3-4ca3-b4e9-dea11aceea83",
+        "concepts": [
+          "complex-numbers"
+        ],
+        "prerequisites": [
+          "numbers", "strings", "function-composition"
+        ],
+        "status": "wip"
+      },
       {
         "slug": "old-annalyns-infiltration",
         "name": "Old Annalyn's Infiltration",

exercises/concept/strange-stopwatch/.docs/hints.md

@@ -0,0 +1,36 @@
+# Hints
+
+Using complex numbers for rotations in 2D can leave things much cleaner and numerically more precise.
+
+## 1. Define a 2D vector rotation function
+
+- [Euler's formula][euler] is your friend here.
+- The inbuilt [`complex(x, y)`][complex] method is more efficient than assigning `x + im*y`.
+- There are methods for retrieving the [real][real] part and [imaginary][imaginary] part of a complex number, or both!
+
+## 2. Define a function to find the angle of the stopwatch's hand
+
+- The inbulit [`angle`][angle] method can be used to quickly find an argument from a complex number.
+- You may want to use your `rotation` function with `angle` because you will need a rotation to redefine the zero point, since `angle` uses the convention of having zero on the negative x-axis.
+- A translation would also be needed when using `angle` since the convention used has angles between -π and π.
+
+## 3. Define a function to tell the time on the stopwatch
+
+- This is a good opportunity to use your `timearg` function.
+- There is an inbuilt method [rad2deg][rad2deg] to convert from radians to degrees.
+- Minutes are the [`abs`][abs] value of the vector.
+
+## 4. Define a function to set a timer
+
+- Consider using the [polar form][euler] of a complex number (e.g. `m*ℯ^(iθ)`).
+- There is an inbuilt method [deg2rad][deg2rad] to convert from degrees to radians.
+- A rotation may be needed to redefine the zero point.
+
+[euler]: https://docs.julialang.org/en/v1/base/math/#Base.cis
+[complex]: https://docs.julialang.org/en/v1/base/numbers/#Base.complex-Tuple{Complex}
+[real]: https://docs.julialang.org/en/v1/base/math/#Base.real
+[imaginary]: https://docs.julialang.org/en/v1/base/math/#Base.imag
+[angle]: https://docs.julialang.org/en/v1/base/math/#Base.angle
+[abs]: https://docs.julialang.org/en/v1/base/math/#Base.abs
+[rad2deg]: https://docs.julialang.org/en/v1/base/math/#Base.Math.rad2deg
+[deg2rad]: https://docs.julialang.org/en/v1/base/math/#Base.Math.deg2rad

exercises/concept/strange-stopwatch/.docs/instructions.md

@@ -0,0 +1,69 @@
+# Instructions
+
+You've been given a strange stopwatch which has no markings on it but takes 60 seconds to go around once.
+What's stranger is that the hand grows by one unit with each passing minute.
+
+To read the time on stopwatch you can place it on some graph paper and read off the x-y coordinates.
+Then we will have to convert these coordinates into minutes and seconds.
+
+On the other hand, we may want to use it as a timer, so we need to be able to put a mark on the paper as a reference time.
+Here we will have to convert the minutes and seconds into the x-y coordinates.
+
+These operations can be done through trigonometric functions and/or rotation matrices, but they can be made simpler (and more fun, I assure you) with the use of complex numbers.
+This ease, which can make rotations quite straightforward, results from Euler's elegant formula, `ℯ^(iθ) = cos(θ) + isin(θ) = x + iy`, where `i = √-1` is the imaginary unit.
+For example, the complex number `z = x + iy`, can be rotated about the origin with a simple multiplication `z * ℯ^(-iθ)`.
+Here the `x` and `y` are just the coordnates on a real 2D plane.
+**Caveat:** A *clockwise* rotation takes a negative exponent, `ℯ^(-iθ) = cos(θ) - isin(θ)`, and a *counterclockwise* rotation takes a positive one, `ℯ^(iθ) = cos(θ) + isin(θ)`.
+
+## 1. Define a 2D vector rotation function
+
+Implement the `rotate` function which takes an x coordinate, a y coordinate and an angle θ (in radians).
+The function should rotate the point about the origin by the given angle θ and return the new coordinates as a tuple.
+While our stopwatch hand will only have integer lengths, this function should rotate a vector of any length.
+
+```julia-repl
+julia> rotate(0, 1, π)
+(0, -1)
+
+julia> rotate(1, 1, π)
+(-1, -1)
+```
+
+## 2. Define a function to find the angle of the stopwatch's hand
+
+Implement the function `timearg` which takes the x-y coordinates the hand and returns the angle in radians.
+Since we are dealing with a clock, the zero point will be on the y-axis (i.e. imaginary axis).
+
+
+```julia-repl
+julia> timearg(0, 1)
+0
+
+julia> timearg(1, 0)
+1.5707963267948966  # equal to π/2
+```
+
+## 3. Define a function to tell the time on the stopwatch
+
+Implement a function `readtime` which takes the x-y coordinates of the hand and returns the time as `"Mm Ss"`.
+
+
+```julia-repl
+julia> readtime(1, 0)
+"1m 15.0s"
+
+julia> readtime(√2, √2)
+"2m 7.5s"
+```
+
+## 4. Define a function to set a timer
+
+Implement a function `getcoords` which takes a time, in minutes and seconds, and outputs the x-y coordinates of the tip of the stopwatch's hand at that time.
+
+```julia-repl
+julia> getcoords(1, 0)
+(0, 1)
+
+julia> getcoords(2, 15)
+(2, 0)
+```

exercises/concept/strange-stopwatch/.meta/config.json

@@ -0,0 +1,17 @@
+{
+  "authors": [
+    "depial"
+  ],
+  "files": {
+    "solution": [
+      "strange-stopwatch.jl"
+    ],
+    "test": [
+      "runtests.jl"
+    ],
+    "exemplar": [
+      ".meta/exemplar.jl"
+    ]
+  },
+  "blurb": "Learn about simple rotations of 2D vectors using complex numbers"
+}

exercises/concept/strange-stopwatch/.meta/design.md

@@ -0,0 +1,26 @@
+# Design
+
+## Goal
+
+The goal of this exercise is to introduce the student to the predefined type of complex numbers.
+
+## Learning objectives
+
+- Understand the predefined type of complex numbers and their construction.
+- Become familiar with a basic subset of functions, such as `complex`, `abs`, `angle`, `reim`, etc.
+
+## Out of scope
+
+- Modular Arithmetic
+
+## Concepts
+
+The Concepts this exercise unlocks are:
+
+- `complex-numbers`
+
+## Prerequisites
+
+- `numbers`
+- `strings`
+- `function-composition`

exercises/concept/strange-stopwatch/.meta/exemplar.jl

@@ -0,0 +1,4 @@
+rotate(x, y, θ) = reim(complex(x, y)cispi(θ/π))
+timearg(x, y) = π - angle(complex(rotate(x, y, π/2)...))
+readtime(x, y) = "$(floor(Int, abs(complex(x, y))))m $(rad2deg(timearg(x, y))/6)s"
+getcoords(m, s) = rotate(reim(m * cispi(-deg2rad(6s)/π))..., π/2)

exercises/concept/strange-stopwatch/runtests.jl

@@ -0,0 +1,61 @@
+using Test
+
+include("strange-stopwatch.jl")
+
+@testset verbose = true "tests" begin
+    @testset "rotations" begin
+        x, y = rotate(1, 0, π/2)
+        @test isapprox(x, 0; atol=1e-2) && isapprox(y, 1; atol=1e-2)
+
+        x, y = rotate(1, 0, -π/2)
+        @test isapprox(x, 0; atol=1e-2) && isapprox(y, -1; atol=1e-2)
+
+        x, y = rotate(1, 1, π)
+        @test isapprox(x, -1; atol=1e-2) && isapprox(y, -1; atol=1e-2)
+
+        x, y = rotate(2, 2, -π/3)
+        @test isapprox(x, 2.73205; atol=1e-2) && isapprox(y, -0.73205; atol=1e-2)
+    end
+
+    @testset "time argument" begin
+        @test isapprox(timearg(0, 1), 0; atol=1e-2)
+
+        @test isapprox(timearg(2, 2), π/4; atol=1e-2)
+
+        @test isapprox(timearg(0, -7), π; atol=1e-2)
+
+        @test isapprox(timearg(-4, 0), 3π/2; atol=1e-2)
+
+        @test isapprox(timearg(1, √3), π/6; atol=1e-2)
+    end
+
+    @testset "read time" begin
+        minsec(strtime) = parse.(Float64, match(r"(\d+\.?\d*)m (\d+\.?\d*)s", strtime).captures)
+
+        m, s = minsec(readtime(1, 0))
+        @test isapprox(m, 1.0; atol=1e-9) && isapprox(s, 15.0; atol=1e-2)
+
+        m, s = minsec(readtime(-4, 0))
+        @test isapprox(m, 4.0; atol=1e-9) && isapprox(s, 45.0; atol=1e-2)
+
+        m, s = minsec(readtime(0, -3))
+        @test isapprox(m, 3.0; atol=1e-9) && isapprox(s, 30.0; atol=1e-2)
+
+        m, s = minsec(readtime(√2, -√2))
+        @test isapprox(m, 2.0; atol=1e-9) && isapprox(s, 22.5; atol=1e-2)
+    end
+
+    @testset "find timer coordinates" begin
+        x, y = getcoords(1, 0)
+        @test isapprox(x, 0; atol=1e-2) && isapprox(y, 1; atol=1e-2)
+
+        x, y = getcoords(2, 15)
+        @test isapprox(x, 2; atol=1e-2) && isapprox(y, 0; atol=1e-2)
+
+        x, y = getcoords(3, 45)
+        @test isapprox(x, -3; atol=1e-2) && isapprox(y, 0; atol=1e-2)
+
+        x, y = getcoords(4, 37.5)
+        @test isapprox(x, -2.8284; atol=1e-2) && isapprox(y, -2.8284; atol=1e-2)
+    end
+end

exercises/concept/strange-stopwatch/strange-stopwatch.jl

@@ -0,0 +1,15 @@
+function rotate(x, y, θ)
+
+end
+
+function timearg(z)
+
+end
+
+function readtime((a, b))
+
+end
+
+function getcoords(m, s)
+
+end