add more tasks

Author: depial

exercises/concept/tracking-turntable/.docs/hints.md

@@ -2,18 +2,28 @@
 
 Using complex numbers for rotations in 2D can leave things much cleaner and numerically more precise.
 
-## 1. Perform a 2D vector rotation
+## 1. Construct a complex number from Cartesian coordinates
 
-- [Euler's formula][euler] is your friend here.
 - The inbuilt [`complex(x, y)`][complex] method is more efficient than assigning `x + im*y`.
+- See the introduction for a review.
+
+## 2. Construct a complex number from Polar coordinates
+
+- There are a few equivalent ways to do this (e.g. `exp`, `cis`, `cispi`). 
+- See the introduction and instructions for further review.
+
+## 3. Perform a 2D vector rotation
+
+- [Euler's formula][euler] is your friend here and you could use your `euler` function.
 - There are methods for retrieving the [real][real] part and [imaginary][imaginary] part of a complex number, or both!
 
-## 2. Perform a radial displacement
+## 4. Perform a radial displacement
 
+- Your `euler` function can be of use, or not...
 - The inbulit [`angle`][angle] method can be used to quickly find an argument from a complex number.
 - The [`abs`][abs] method can be used to find the magnitude of a complex number.
 
-## 3. Find desired song
+## 5. Find the desired song
 
 - This is a good opportunity to compose your `rotate` and `rdisplace` functions.
 

exercises/concept/tracking-turntable/.docs/instructions.md

@@ -14,15 +14,44 @@ Turndit needs to know how to find the new `(x, y)` coordinates to which the need
 
 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 via rotations and radial displacements.
 
-This ease results from Euler's elegant formula, `ℯ^(iθ) = cos(θ) + isin(θ) = x + iy`, where `i = √-1` is the imaginary unit.
+This ease results from Euler's elegant formula, `ℯ^(iθ) = cos(θ) + isin(θ) = x + iy`, where `i = √-1` is the imaginary unit and `|x + iy| = 1`.
+With `r = |x + iy|`, we have the more general polar form of `r * ℯ^(iθ) = r * (cos(θ) + isin(θ)) = x + iy`.
+
 
 For rotations, the complex number `z = x + iy`, can be rotated an angle `θ` about the origin with a simple multiplication: `z * ℯ^(iθ)`.
 Note that the `x` and `y` here are just the usual coordinates on the real 2D Cartesian plane, and a positive angle results in a *counterclockwise* rotation, while a negative angle results in a *clockwise* one.
 
 Likewise simply, a radial displacement `Δr` can be made by adding it to the magnitude `r` of a complex number in the polar form (eg. `z = r * ℯ^(iθ)` -> `z' = (r + Δr) * ℯ^(iθ)`).
 Note how the angular part stays the same and only the magnitude, `r`, is varied, as expected.
 
-## 1. Perform a 2D vector rotation
+## 1. Construct a complex number from Cartesian coordinates
+
+Implement the `z(x, y)` function which takes an `x` coordinate, a `y` coordinate from the complex plane and returns the equivalent complex number.
+
+```julia-repl
+julia> z(1, 1)
+1.0 + 1.0im
+
+julia> z(4.5, -7.3)
+4.5 - 7.3im
+```
+
+## 2. Construct a complex number from Polar coordinates
+
+Implement the `euler(r, θ)` function, which takes a radial coordinate `r`, an angle `θ` (in radians) and returns the equivalent complex number in rectangular form.
+
+```julia-repl
+julia> euler(1, π)
+-1.0 + 0.0im
+
+julia> euler(3, π/2)
+0.0 + 3.0im
+
+julia> euler(2, -π/4)
+1.4142135623730951 - 1.414213562373095im  # √2 - √2im
+```
+
+## 3. Perform a 2D vector rotation
 
 Implement the `rotate(x, y, θ)` 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.
@@ -34,7 +63,7 @@ julia> rotate(0, 1, π)
 julia> rotate(1, 1, -π/2)
 (1, -1)
 ```
-## 2. Perform a radial displacement
+## 4. Perform a radial displacement
 
 Implement the function `rdisplace(x, y, r)` which takes an `x` coordinate, a `y` coordinate and a radial displacement `r`.
 The function should displace the point along the radius by the amount `r` and return the new coordinates as a tuple.
@@ -46,7 +75,7 @@ julia> rdisplace(0, 1, 1)
 julia> rdisplace(1, 1, √2)
 (2, 2)
 ```
-## 3. Find desired song
+## 5. Find the desired song
 
 Implement a function `findsong(x, y, r, θ)` which takes the x and y coordinates of the needle as well as the radial and angular displacement between the needle and the beginning of the desired song. The new coordinates should be returned as a tuple.
 

exercises/concept/tracking-turntable/.meta/exemplar.jl

@@ -1,3 +1,5 @@
-rotate(x, y, θ) = reim(complex(x, y)cispi(θ/π))
-rdisplace(x, y, r) = reim((abs(complex(x, y)) + r)cispi(angle(complex(x, y)) / π))
+z = complex
+euler(r, θ) = r * cispi(θ / π)
+rotate(x, y, θ) = reim(z(x, y)euler(1, θ))
+rdisplace(x, y, r) = reim((abs(z(x, y)) + r)cispi(angle(z(x, y)) / π))
 findsong(x, y, r, θ) = rotate(rdisplace(x, y, r)..., θ)

exercises/concept/tracking-turntable/runtests.jl

@@ -1,8 +1,27 @@
 using Test
 
-include("tracking-turntable.jl")
+#include("tracking-turntable.jl")
+include(".meta/exemplar.jl")
 
 @testset verbose = true "tests" begin
+    @testset "complex from cartesian" begin
+        @test z(1, 1) ≈ 1 + 1im
+
+        @test z(4.5, -7.3) ≈ 4.5 - 7.3im
+
+        @test z(-π, 42.24) ≈ -π + 42.24im
+    end
+    
+    @testset "complex from polar" begin
+        @test euler(1, π) ≈ -1.0 + 0.0im
+
+        @test euler(3, π/2) ≈ 0.0 + 3.0im
+
+        @test isapprox(euler(2, -π/4), √2 - √2im; atol=1e-2)
+
+        @test isapprox(euler(2.5, -4π/3), -1.25 + 2.165im; atol=1e-2)
+    end
+
     @testset "rotations" begin
         x, y = rotate(1, 0, π/2)
         @test isapprox(x, 0.0; atol=1e-2) && isapprox(y, 1.0; atol=1e-2)