Maths

For all the power of modern computers to perform huge sums at lightning speed, the average developer rarely uses any mathematics to do their job. But not today! Today we'll use mathematics to solve a real problem. And not boring mathematics - we're going to use trigonometry and vectors and all sorts of stuff that you always said you'd never have to use after highschool.
The Problem
You want to make an SVG of a clock. Not a digital clock - no, that would be easy - an analogue clock, with hands. You're not looking for anything fancy, just a nice function that takes a Time from the time package and spits out an SVG of a clock with all the hands - hour, minute and second - pointing in the right direction. How hard can that be?
First we're going to need an SVG of a clock for us to play with. SVGs are a fantastic image format to manipulate programmatically because they're written as a series of shapes, described in XML. So this clock:
an svg of a clock
is described like this:
<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg xmlns="http://www.w3.org/2000/svg"
     width="100%"
     height="100%"
     viewBox="0 0 300 300"
     version="2.0">
​
  <!-- bezel -->
  <circle cx="150" cy="150" r="100" style="fill:#fff;stroke:#000;stroke-width:5px;"/>
​
  <!-- hour hand -->
  <line x1="150" y1="150" x2="114.150000" y2="132.260000"
        style="fill:none;stroke:#000;stroke-width:7px;"/>
​
  <!-- minute hand -->
  <line x1="150" y1="150" x2="101.290000" y2="99.730000"
        style="fill:none;stroke:#000;stroke-width:7px;"/>
​
  <!-- second hand -->
  <line x1="150" y1="150" x2="77.190000" y2="202.900000"
        style="fill:none;stroke:#f00;stroke-width:3px;"/>
</svg>
It's a circle with three lines, each of the lines starting in the middle of the circle (x=150, y=150), and ending some distance away.
So what we're going to do is reconstruct the above somehow, but change the lines so they point in the appropriate directions for a given time.
An Acceptance Test
Before we get too stuck in, lets think about an acceptance test. We've got an example clock, so let's think about what the important parameters are going to be.
<line x1="150" y1="150" x2="114.150000" y2="132.260000"
        style="fill:none;stroke:#000;stroke-width:7px;"/>
The centre of the clock (the attributes x1 and y1 for this line) is the same for each hand of the clock. The numbers that need to change for each hand of the clock - the parameters to whatever builds the SVG - are the x2 and y2 attributes. We'll need an X and a Y for each of the hands of the clock.
I could think about more parameters - the radius of the clockface circle, the size of the SVG, the colours of the hands, their shape, etc... but it's better to start off by solving a simple, concrete problem with a simple, concrete solution, and then to start adding parameters to make it generalised.
So we'll say that
every clock has a centre of (150, 150)
the hour hand is 50 long
the minute hand is 80 long
the second hand is 90 long.
A thing to note about SVGs: the origin - point (0,0) - is at the top left hand corner, not the bottom left as we might expect. It'll be important to remember this when we're working out where what numbers to plug in to our lines.
Finally, I'm not deciding how to construct the SVG - we could use a template from the text/template package, or we could just send bytes into a bytes.Buffer or a writer. But we know we'll need those numbers, so let's focus on testing something that creates them.
Write the test first
So my first test looks like this:
package clockface_test
​
import (
    "testing"
    "time"
​
    "github.com/gypsydave5/learn-go-with-tests/math/v1/clockface"
)
​
func TestSecondHandAtMidnight(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 0, 0, time.UTC)
​
    want := clockface.Point{X: 150, Y: 150 - 90}
    got := clockface.SecondHand(tm)
​
    if got != want {
        t.Errorf("Got %v, wanted %v", got, want)
    }
}
Remember how SVGs plot their coordinates from the top left hand corner? To place the second hand at midnight we expect that it hasn't moved from the centre of the clockface on the X axis - still 150 - and the Y axis is the length of the hand 'up' from the centre; 150 minus 90.
Try to run the test
This drives out the expected failures around the missing functions and types:
--- FAIL: TestSecondHandAtMidnight (0.00s)
# github.com/gypsydave5/learn-go-with-tests/math/v1/clockface_test [github.com/gypsydave5/learn-go-with-tests/math/v1/clockface.test]
./clockface_test.go:13:10: undefined: clockface.Point
./clockface_test.go:14:9: undefined: clockface.SecondHand
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v1/clockface [build failed]
So a Point where the tip of the second hand should go, and a function to get it.
Write the minimal amount of code for the test to run and check the failing test output
Let's implement those types to get the code to compile
package clockface
​
import "time"
​
// A Point represents a two dimensional Cartesian coordinate
type Point struct {
    X float64
    Y float64
}
​
// SecondHand is the unit vector of the second hand of an analogue clock at time `t`
// represented as a Point.
func SecondHand(t time.Time) Point {
    return Point{}
}
and now we get
--- FAIL: TestSecondHandAtMidnight (0.00s)
    clockface_test.go:17: Got {0 0}, wanted {150 60}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v1/clockface    0.006s
Write enough code to make it pass
When we get the expected failure, we can fill in the return value of HandsAt:
// SecondHand is the unit vector of the second hand of an analogue clock at time `t`
// represented as a Point.
func SecondHand(t time.Time) Point {
    return Point{150, 60}
}
Behold, a passing test.
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v1/clockface    0.006s
Refactor
No need to refactor yet - there's barely enough code!
Repeat for new requirements
We probably need to do some work here that doesn't just involve returning a clock that shows midnight for every time...
Write the test first
func TestSecondHandAt30Seconds(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 30, 0, time.UTC)
​
    want := clockface.Point{X: 150, Y: 150 + 90}
    got := clockface.SecondHand(tm)
​
    if got != want {
        t.Errorf("Got %v, wanted %v", got, want)
    }
}
Same idea, but now the second hand is pointing downwards so we add the length to the Y axis.
This will compile... but how do we make it pass?
Thinking time
How are we going to solve this problem?
Every minute the second hand goes through the same 60 states, pointing in 60 different directions. When it's 0 seconds it points to the top of the clockface, when it's 30 seconds it points to the bottom of the clockface. Easy enough.
So if I wanted to think about in what direction the second hand was pointing at, say, 37 seconds, I'd want the angle between 12 o'clock and 37/60ths around the circle. In degrees this is (360 / 60 ) * 37 = 222, but it's easier just to remember that it's 37/60 of a complete rotation.
But the angle is only half the story; we need to know the X and Y coordinate that the tip of the second hand is pointing at. How can we work that out?
Math
Imagine a circle with a radius of 1 drawn around the origin - the coordinate 0, 0.
picture of the unit circle
This is called the 'unit circle' because... well, the radius is 1 unit!
The circumference of the circle is made of points on the grid - more coordinates. The x and y components of each of these coordinates form a triangle, the hypotenuse of which is always 1 - the radius of the circle
picture of the unit circle with a point defined on the circumference
Now, trigonometry will let us work out the lengths of X and Y for each triangle if we know the angle they make with the origin. The X coordinate will be cos(a), and the Y coordinate will be sin(a), where a is the angle made between the line and the (positive) x axis.
picture of the unit circle with the x and y elements of a ray defined as cos(a) and sin(a) respectively, where a is the angle made by the ray with the x axis
(If you don't believe this, go and look at Wikipedia...)
One final twist - because we want to measure the angle from 12 o'clock rather than from the X axis (3 o'clock), we need to swap the axis around; now x = sin(a) and y = cos(a).
unit circle ray defined from by angle from y axis
So now we know how to get the angle of the second hand (1/60th of a circle for each second) and the X and Y coordinates. We'll need functions for both sin and cos.
math
Happily the Go math package has both, with one small snag we'll need to get our heads around; if we look at the description of math.Cos:
Cos returns the cosine of the radian argument x.
It wants the angle to be in radians. So what's a radian? Instead of defining the full turn of a circle to be made up of 360 degrees, we define a full turn as being 2π radians. There are good reasons to do this that we won't go in to.
Now that we've done some reading, some learning and some thinking, we can write our next test.
Write the test first
All this maths is hard and confusing. I'm not confident I understand what's going on - so let's write a test! We don't need to solve the whole problem in one go - let's start off with working out the correct angle, in radians, for the second hand at a particular time.
I'm going to write these tests within the clockface package; they may never get exported, and they may get deleted (or moved) once I have a better grip on what's going on.
I'm also going to comment out the acceptance test that I was working on while I'm working on these tests - I don't want to get distracted by that test while I'm getting this one to pass.
package clockface
​
import (
    "math"
    "testing"
    "time"
)
​
func TestSecondsInRadians(t *testing.T) {
    thirtySeconds := time.Date(312, time.October, 28, 0, 0, 30, 0, time.UTC)
    want := math.Pi
    got := secondsInRadians(thirtySeconds)
​
    if want != got {
        t.Fatalf("Wanted %v radians, but got %v", want, got)
    }
}
Here we're testing that 30 seconds past the minute should put the second hand at halfway around the clock. And it's our first use of the math package! If a full turn of a circle is 2π radians, we know that halfway round should just be π radians. math.Pi provides us with a value for π.
Try to run the test
# github.com/gypsydave5/learn-go-with-tests/math/v2/clockface [github.com/gypsydave5/learn-go-with-tests/math/v2/clockface.test]
./clockface_test.go:12:9: undefined: secondsInRadians
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v2/clockface [build failed]
Write the minimal amount of code for the test to run and check the failing test output
func secondsInRadians(t time.Time) float64 {
    return 0
}
--- FAIL: TestSecondsInRadians (0.00s)
    clockface_test.go:15: Wanted 3.141592653589793 radians, but got 0
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v2/clockface    0.007s
Write enough code to make it pass
func secondsInRadians(t time.Time) float64 {
    return math.Pi
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v2/clockface    0.011s
Refactor
Nothing needs refactoring yet
Repeat for new requirements
Now we can extend the test to cover a few more scenarios. I'm going to skip forward a bit and show some already refactored test code - it should be clear enough how I got where I want to.
func TestSecondsInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(0, 0, 30), math.Pi},
        {simpleTime(0, 0, 0), 0},
        {simpleTime(0, 0, 45), (math.Pi / 2) * 3},
        {simpleTime(0, 0, 7), (math.Pi / 30) * 7},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := secondsInRadians(c.time)
            if got != c.angle {
                t.Fatalf("Wanted %v radians, but got %v", c.angle, got)
            }
        })
    }
}
I added a couple of helper functions to make writing this table based test a little less tedious. testName converts a time into a digital watch format (HH:MM:SS), and simpleTime constructs a time.Time using only the parts we actually care about (again, hours, minutes and seconds).
func simpleTime(hours, minutes, seconds int) time.Time {
    return time.Date(312, time.October, 28, hours, minutes, seconds, 0, time.UTC)
}
​
func testName(t time.Time) string {
    return t.Format("15:04:05")
}
These two functions should help make these tests (and future tests) a little easier to write and maintain.
This gives us some nice test output:
--- FAIL: TestSecondsInRadians (0.00s)
    --- FAIL: TestSecondsInRadians/00:00:00 (0.00s)
        clockface_test.go:24: Wanted 0 radians, but got 3.141592653589793
    --- FAIL: TestSecondsInRadians/00:00:45 (0.00s)
        clockface_test.go:24: Wanted 4.71238898038469 radians, but got 3.141592653589793
    --- FAIL: TestSecondsInRadians/00:00:07 (0.00s)
        clockface_test.go:24: Wanted 0.7330382858376184 radians, but got 3.141592653589793
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v3/clockface    0.007s
Time to implement all of that maths stuff we were talking about above:
func secondsInRadians(t time.Time) float64 {
    return float64(t.Second()) * (math.Pi / 30)
}
One second is (2π / 60) radians... cancel out the 2 and we get π/30 radians. Multiply that by the number of seconds (as a float64) and we should now have all the tests passing...
--- FAIL: TestSecondsInRadians (0.00s)
    --- FAIL: TestSecondsInRadians/00:00:30 (0.00s)
        clockface_test.go:24: Wanted 3.141592653589793 radians, but got 3.1415926535897936
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v3/clockface    0.006s
Wait, what?
Floats are horrible
Floating point arithmetic is notoriously inaccurate. Computers can only really handle integers, and rational numbers to some extent. Decimal numbers start to become inaccurate, especially when we factor them up and down as we are in the secondsInRadians function. By dividing math.Pi by 30 and then by multiplying it by 30 we've ended up with a number that's no longer the same as math.Pi.
There are two ways around this:
Live with the it
Refactor our function by refactoring our equation
Now (1) may not seem all that appealing, but it's often the only way to make floating point equality work. Being inaccurate by some infinitesimal fraction is frankly not going to matter for the purposes of drawing a clockface, so we could write a function that defines a 'close enough' equality for our angles. But there's a simple way we can get the accuracy back: we rearrange the equation so that we're no longer dividing down and then multiplying up. We can do it all by just dividing.
So instead of
numberOfSeconds * π / 30
we can write
π / (30 / numberOfSeconds)
which is equivalent.
In Go:
func secondsInRadians(t time.Time) float64 {
    return (math.Pi / (30 / (float64(t.Second()))))
}
And we get a pass.
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v2/clockface     0.005s
Repeat for new requirements
So we've got the first part covered here - we know what angle the second hand will be pointing at in radians. Now we need to work out the coordinates.
Again, let's keep this as simple as possible and only work with the unit circle; the circle with a radius of 1. This means that our hands will all have a length of one but, on the bright side, it means that the maths will be easy for us to swallow.
Write the test first
func TestSecondHandVector(t *testing.T) {
    cases := []struct {
        time  time.Time
        point Point
    }{
        {simpleTime(0, 0, 30), Point{0, -1}},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := secondHandPoint(c.time)
            if got != c.point {
                t.Fatalf("Wanted %v Point, but got %v", c.point, got)
            }
        })
    }
}
Try to run the test
# github.com/gypsydave5/learn-go-with-tests/math/v4/clockface [github.com/gypsydave5/learn-go-with-tests/math/v4/clockface.test]
./clockface_test.go:40:11: undefined: secondHandPoint
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v4/clockface [build failed]
Write the minimal amount of code for the test to run and check the failing test output
func secondHandPoint(t time.Time) Point {
    return Point{}
}
--- FAIL: TestSecondHandPoint (0.00s)
    --- FAIL: TestSecondHandPoint/00:00:30 (0.00s)
        clockface_test.go:42: Wanted {0 -1} Point, but got {0 0}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v4/clockface    0.010s
Write enough code to make it pass
func secondHandPoint(t time.Time) Point {
    return Point{0, -1}
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v4/clockface    0.007s
Repeat for new requirements
func TestSecondHandPoint(t *testing.T) {
    cases := []struct {
        time  time.Time
        point Point
    }{
        {simpleTime(0, 0, 30), Point{0, -1}},
        {simpleTime(0, 0, 45), Point{-1, 0}},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := secondHandPoint(c.time)
            if got != c.point {
                t.Fatalf("Wanted %v Point, but got %v", c.point, got)
            }
        })
    }
}
Try to run the test
--- FAIL: TestSecondHandPoint (0.00s)
    --- FAIL: TestSecondHandPoint/00:00:45 (0.00s)
        clockface_test.go:43: Wanted {-1 0} Point, but got {0 -1}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v4/clockface    0.006s
Write enough code to make it pass
Remember our unit circle picture?
picture of the unit circle with the x and y elements of a ray defined as cos(a) and sin(a) respectively, where a is the angle made by the ray with the x axis
We now want the equation that produces X and Y. Let's write it into seconds:
func secondHandPoint(t time.Time) Point {
    angle := secondsInRadians(t)
    x := math.Sin(angle)
    y := math.Cos(angle)
​
    return Point{x, y}
}
Now we get
--- FAIL: TestSecondHandPoint (0.00s)
    --- FAIL: TestSecondHandPoint/00:00:30 (0.00s)
        clockface_test.go:43: Wanted {0 -1} Point, but got {1.2246467991473515e-16 -1}
    --- FAIL: TestSecondHandPoint/00:00:45 (0.00s)
        clockface_test.go:43: Wanted {-1 0} Point, but got {-1 -1.8369701987210272e-16}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v4/clockface    0.007s
Wait, what (again)? Looks like we've been cursed by the floats once more - both of those unexpected numbers are infinitesimal - way down at the 16th decimal place. So again we can either choose to try to increase precision, or to just say that they're roughly equal and get on with our lives.
One option to increase the accuracy of these angles would be to use the rational type Rat from the math/big package. But given the objective is to draw an SVG and not the moon landings I think we can live with a bit of fuzziness.
func TestSecondHandPoint(t *testing.T) {
    cases := []struct {
        time  time.Time
        point Point
    }{
        {simpleTime(0, 0, 30), Point{0, -1}},
        {simpleTime(0, 0, 45), Point{-1, 0}},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := secondHandPoint(c.time)
            if !roughlyEqualPoint(got, c.point) {
                t.Fatalf("Wanted %v Point, but got %v", c.point, got)
            }
        })
    }
}
​
func roughlyEqualFloat64(a, b float64) bool {
    const equalityThreshold = 1e-7
    return math.Abs(a-b) < equalityThreshold
}
​
func roughlyEqualPoint(a, b Point) bool {
    return roughlyEqualFloat64(a.X, b.X) &&
        roughlyEqualFloat64(a.Y, b.Y)
}
We've defined two functions to define approximate equality between two Points
they'll work if the X and Y elements are within 0.0000001 of each other.
That's still pretty accurate.
and now we get
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v4/clockface    0.007s
Refactor
I'm still pretty happy with this.
Repeat for new requirements
Well, saying new isn't entirely accurate - really what we can do now is get that acceptance test passing! Let's remind ourselves of what it looks like:
func TestSecondHandAt30Seconds(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 30, 0, time.UTC)
​
    want := clockface.Point{X: 150, Y: 150 + 90}
    got := clockface.SecondHand(tm)
​
    if got != want {
        t.Errorf("Got %v, wanted %v", got, want)
    }
}
Try to run the test
--- FAIL: TestSecondHandAt30Seconds (0.00s)
    clockface_acceptance_test.go:28: Got {150 60}, wanted {150 240}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v5/clockface    0.007s
Write enough code to make it pass
We need to do three things to convert our unit vector into a point on the SVG:
Scale it to the length of the hand
Flip it over the X axis because to account for the SVG having an origin in
the top left hand corner
Translate it to the right position (so that it's coming from an origin of
(150,150))
Fun times!
// SecondHand is the unit vector of the second hand of an analogue clock at time `t`
// represented as a Point.
func SecondHand(t time.Time) Point {
    p := secondHandPoint(t)
    p = Point{p.X * 90, p.Y * 90}   // scale
    p = Point{p.X, -p.Y}            // flip
    p = Point{p.X + 150, p.Y + 150} // translate
    return p
}
Scale, flip, and translated in exactly that order. Hooray maths!
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v5/clockface    0.007s
Refactor
There's a few magic numbers here that should get pulled out as constants, so let's do that
const secondHandLength = 90
const clockCentreX = 150
const clockCentreY = 150
​
// SecondHand is the unit vector of the second hand of an analogue clock at time `t`
// represented as a Point.
func SecondHand(t time.Time) Point {
    p := secondHandPoint(t)
    p = Point{p.X * secondHandLength, p.Y * secondHandLength}
    p = Point{p.X, -p.Y}
    p = Point{p.X + clockCentreX, p.Y + clockCentreY} //translate
    return p
}
Draw the clock
Well... the second hand anyway...
Let's do this thing - because there's nothing worse than not delivering some value when it's just sitting there waiting to get out into the world to dazzle people. Let's draw a second hand!
We're going to stick a new directory under our main clockface package directory, called (confusingly), clockface. In there we'll put the main package that will create the binary that will build an SVG:
├── clockface
│   └── main.go
├── clockface.go
├── clockface_acceptance_test.go
└── clockface_test.go
and inside main.go
package main
​
import (
    "fmt"
    "io"
    "os"
    "time"
​
    "github.com/gypsydave5/learn-go-with-tests/math/v6/clockface"
)
​
func main() {
    t := time.Now()
    sh := clockface.SecondHand(t)
    io.WriteString(os.Stdout, svgStart)
    io.WriteString(os.Stdout, bezel)
    io.WriteString(os.Stdout, secondHandTag(sh))
    io.WriteString(os.Stdout, svgEnd)
}
​
func secondHandTag(p clockface.Point) string {
    return fmt.Sprintf(`<line x1="150" y1="150" x2="%f" y2="%f" style="fill:none;stroke:#f00;stroke-width:3px;"/>`, p.X, p.Y)
}
​
const svgStart = `<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg xmlns="http://www.w3.org/2000/svg"
     width="100%"
     height="100%"
     viewBox="0 0 300 300"
     version="2.0">`
​
const bezel = `<circle cx="150" cy="150" r="100" style="fill:#fff;stroke:#000;stroke-width:5px;"/>`
​
const svgEnd = `</svg>`
Oh boy am I not trying to win any prizes for beautiful code with this mess - but it does the job. It's writing an SVG out to os.Stdout - one string at a time.
If we build this
go build
and run it, sending the output into a file
./clockface > clock.svg
We should see something like
a clock with only a second hand
Refactor
This stinks. Well, it doesn't quite stink stink, but I'm not happy about it.
That whole SecondHand function is super tied to being an SVG... without
mentioning SVGs or actually producing an SVG...
... while at the same time I'm not testing any of my SVG code.
Yeah, I guess I screwed up. This feels wrong. Let's try and recover with a more SVG-centric test.
What are our options? Well, we could try testing that the characters spewing out of the SVGWriter contain things that look like the sort of SVG tag we're expecting for a particular time. For instance:
func TestSVGWriterAtMidnight(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 0, 0, time.UTC)
​
    var b strings.Builder
    clockface.SVGWriter(&b, tm)
    got := b.String()
​
    want := `<line x1="150" y1="150" x2="150" y2="60"`
​
    if !strings.Contains(got, want) {
        t.Errorf("Expected to find the second hand %v, in the SVG output %v", want, got)
    }
}
But is this really an improvement?
Not only will it still pass if I don't produce a valid SVG (as it's only testing that a string appears in the output), but it will also fail if I make the smallest, unimportant change to that string - if I add an extra space between the attributes, for instance.
The biggest smell is really that I'm testing a data structure - XML - by looking at its representation as a series of characters - as a string. This is never, ever a good idea as it produces problems just like the ones I outline above: a test that's both too fragile and not sensitive enough. A test that's testing the wrong thing!
So the only solution is to test the output as XML. And to do that we'll need to parse it.
Parsing XML
​encoding/xml is the Go package that can handle all things to do with simple XML parsing.
The function xml.Unmarshall takes a []byte of XML data and a pointer to a struct for it to get unmarshalled in to.
So we'll need a struct to unmarshall our XML into. We could spend some time working out what the correct names for all of the nodes and attributes, and how to write the correct structure but, happily, someone has written zek a program that will automate all of that hard work for us. Even better, there's an online version at https://www.onlinetool.io/xmltogo/. Just paste the SVG from the top of the file into one box and - bam
out pops:
type Svg struct {
    XMLName xml.Name `xml:"svg"`
    Text    string   `xml:",chardata"`
    Xmlns   string   `xml:"xmlns,attr"`
    Width   string   `xml:"width,attr"`
    Height  string   `xml:"height,attr"`
    ViewBox string   `xml:"viewBox,attr"`
    Version string   `xml:"version,attr"`
    Circle  struct {
        Text  string `xml:",chardata"`
        Cx    string `xml:"cx,attr"`
        Cy    string `xml:"cy,attr"`
        R     string `xml:"r,attr"`
        Style string `xml:"style,attr"`
    } `xml:"circle"`
    Line []struct {
        Text  string `xml:",chardata"`
        X1    string `xml:"x1,attr"`
        Y1    string `xml:"y1,attr"`
        X2    string `xml:"x2,attr"`
        Y2    string `xml:"y2,attr"`
        Style string `xml:"style,attr"`
    } `xml:"line"`
}
We could make adjustments to this if we needed to (like changing the name of the struct to SVG) but it's definitely good enough to start us off.
func TestSVGWriterAtMidnight(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 0, 0, time.UTC)
​
    b := bytes.Buffer{}
    clockface.SVGWriter(&b, tm)
​
    svg := Svg{}
    xml.Unmarshal(b.Bytes(), &svg)
​
    x2 := "150"
    y2 := "60"
​
    for _, line := range svg.Line {
        if line.X2 == x2 && line.Y2 == y2 {
            return
        }
    }
​
    t.Errorf("Expected to find the second hand with x2 of %+v and y2 of %+v, in the SVG output %v", x2, y2, b.String())
}
We write the output of clockface.SVGWriter to a bytes.Buffer and then Unmarshall it into an Svg. We then look at each Line in the Svg to see if any of them have the expected X2 and Y2 values. If we get a match we return early (passing the test); if not we fail with a (hopefully) informative message.
# github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface_test [github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface.test]
./clockface_acceptance_test.go:41:2: undefined: clockface.SVGWriter
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface [build failed]
Looks like we'd better write that SVGWriter...
package clockface
​
import (
    "fmt"
    "io"
    "time"
)
​
const (
    secondHandLength = 90
    clockCentreX     = 150
    clockCentreY     = 150
)
​
//SVGWriter writes an SVG representation of an analogue clock, showing the time t, to the writer w
func SVGWriter(w io.Writer, t time.Time) {
    io.WriteString(w, svgStart)
    io.WriteString(w, bezel)
    secondHand(w, t)
    io.WriteString(w, svgEnd)
}
​
func secondHand(w io.Writer, t time.Time) {
    p := secondHandPoint(t)
    p = Point{p.X * secondHandLength, p.Y * secondHandLength} // scale
    p = Point{p.X, -p.Y}                                      // flip
    p = Point{p.X + clockCentreX, p.Y + clockCentreY}         // translate
    fmt.Fprintf(w, `<line x1="150" y1="150" x2="%.3f" y2="%.3f" style="fill:none;stroke:#f00;stroke-width:3px;"/>`, p.X, p.Y)
}
​
const svgStart = `<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg xmlns="http://www.w3.org/2000/svg"
     width="100%"
     height="100%"
     viewBox="0 0 300 300"
     version="2.0">`
​
const bezel = `<circle cx="150" cy="150" r="100" style="fill:#fff;stroke:#000;stroke-width:5px;"/>`
​
const svgEnd = `</svg>`
The most beautiful SVG writer? No. But hopefully it'll do the job...
--- FAIL: TestSVGWriterAtMidnight (0.00s)
    clockface_acceptance_test.go:56: Expected to find the second hand with x2 of 150 and y2 of 60, in the SVG output <?xml version="1.0" encoding="UTF-8" standalone="no"?>
        <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
        <svg xmlns="http://www.w3.org/2000/svg"
             width="100%"
             height="100%"
             viewBox="0 0 300 300"
             version="2.0"><circle cx="150" cy="150" r="100" style="fill:#fff;stroke:#000;stroke-width:5px;"/><line x1="150" y1="150" x2="150.000000" y2="60.000000" style="fill:none;stroke:#f00;stroke-width:3px;"/></svg>
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface    0.008s
Oooops! The %f format directive is printing our coordinates to the default level of precision - six decimal places. We should be explicit as to what level of precision we're expecting for the coordinates. Let's say three decimal places.
s := fmt.Sprintf(`<line x1="150" y1="150" x2="%.3f" y2="%.3f" style="fill:none;stroke:#f00;stroke-width:3px;"/>`, p.X, p.Y)
And after we update our expectations in the test
    x2 := "150.000"
    y2 := "60.000"
We get:
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface    0.006s
We can now shorten our main function:
package main
​
import (
    "os"
    "time"
​
    "github.com/gypsydave5/learn-go-with-tests/math/v7b/clockface"
)
​
func main() {
    t := time.Now()
    clockface.SVGWriter(os.Stdout, t)
}
And we can write a test for another time following the same pattern, but not before...
Refactor
Three things stick out:
We're not really testing for all of the information we need to ensure is
present - what about the x1 values, for instance?
Also, those attributes for x1 etc. aren't really strings are they? They're
numbers!
Do I really care about the style of the hand? Or, for that matter, the
empty Text node that's been generated by zak?
We can do better. Let's make a few adjustments to the Svg struct, and the tests, to sharpen everything up.
type SVG struct {
    XMLName xml.Name `xml:"svg"`
    Xmlns   string   `xml:"xmlns,attr"`
    Width   float64  `xml:"width,attr"`
    Height  float64  `xml:"height,attr"`
    ViewBox string   `xml:"viewBox,attr"`
    Version string   `xml:"version,attr"`
    Circle  Circle   `xml:"circle"`
    Line    []Line   `xml:"line"`
}
​
type Circle struct {
    Cx float64 `xml:"cx,attr"`
    Cy float64 `xml:"cy,attr"`
    R  float64 `xml:"r,attr"`
}
​
type Line struct {
    X1 float64 `xml:"x1,attr"`
    Y1 float64 `xml:"y1,attr"`
    X2 float64 `xml:"x2,attr"`
    Y2 float64 `xml:"y2,attr"`
}
Here I've
Made the important parts of the struct named types -- the Line and the
Circle
Turned the numeric attributes into float64s instead of strings.
Deleted unused attributes like Style and Text
Renamed Svg to SVG because it's the right thing to do.
This will let us assert more precisely on the line we're looking for:
func TestSVGWriterAtMidnight(t *testing.T) {
    tm := time.Date(1337, time.January, 1, 0, 0, 0, 0, time.UTC)
    b := bytes.Buffer{}
​
    clockface.SVGWriter(&b, tm)
​
    svg := SVG{}
​
    xml.Unmarshal(b.Bytes(), &svg)
​
    want := Line{150, 150, 150, 60}
​
    for _, line := range svg.Line {
        if line == want {
            return
        }
    }
​
    t.Errorf("Expected to find the second hand line %+v, in the SVG lines %+v", want, svg.Line)
}
Finally we can take a leaf out of the unit tests' tables, and we can write a helper function containsLine(line Line, lines []Line) bool to really make these tests shine:
func TestSVGWriterSecondHand(t *testing.T) {
    cases := []struct {
        time time.Time
        line Line
    }{
        {
            simpleTime(0, 0, 0),
            Line{150, 150, 150, 60},
        },
        {
            simpleTime(0, 0, 30),
            Line{150, 150, 150, 240},
        },
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            b := bytes.Buffer{}
            clockface.SVGWriter(&b, c.time)
​
            svg := SVG{}
            xml.Unmarshal(b.Bytes(), &svg)
​
            if !containsLine(c.line, svg.Line) {
                t.Errorf("Expected to find the second hand line %+v, in the SVG lines %+v", c.line, svg.Line)
            }
        })
    }
}
Now that's what I call an acceptance test!
Write the test first
So that's the second hand done. Now let's get started on the minute hand.
func TestSVGWriterMinutedHand(t *testing.T) {
    cases := []struct {
        time time.Time
        line Line
    }{
        {
            simpleTime(0, 0, 0),
            Line{150, 150, 150, 70},
        },
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            b := bytes.Buffer{}
            clockface.SVGWriter(&b, c.time)
​
            svg := SVG{}
            xml.Unmarshal(b.Bytes(), &svg)
​
            if !containsLine(c.line, svg.Line) {
                t.Errorf("Expected to find the minute hand line %+v, in the SVG lines %+v", c.line, svg.Line)
            }
        })
    }
}
Try to run the test
--- FAIL: TestSVGWriterMinutedHand (0.00s)
    --- FAIL: TestSVGWriterMinutedHand/00:00:00 (0.00s)
        clockface_acceptance_test.go:87: Expected to find the minute hand line {X1:150 Y1:150 X2:150 Y2:70}, in the SVG lines [{X1:150 Y1:150 X2:150 Y2:60}]
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v8/clockface    0.007s
We'd better start building some other clockhands, Much in the same way as we produced the tests for the second hand, we can iterate to produce the following set of tests. Again we'll comment out our acceptance test while we get this working:
func TestMinutesInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(0, 30, 0), math.Pi},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := minutesInRadians(c.time)
            if got != c.angle {
                t.Fatalf("Wanted %v radians, but got %v", c.angle, got)
            }
        })
    }
}
Try to run the test
# github.com/gypsydave5/learn-go-with-tests/math/v8/clockface [github.com/gypsydave5/learn-go-with-tests/math/v8/clockface.test]
./clockface_test.go:59:11: undefined: minutesInRadians
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v8/clockface [build failed]
Write the minimal amount of code for the test to run and check the failing test output
func minutesInRadians(t time.Time) float64 {
    return math.Pi
}
Repeat for new requirements
Well, OK - now let's make ourselves do some real work. We could model the minute hand as only moving every full minute - so that it 'jumps' from 30 to 31 minutes past without moving in between. But that would look a bit rubbish. What we want it to do is move a tiny little bit every second.
func TestMinutesInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(0, 30, 0), math.Pi},
        {simpleTime(0, 0, 7), 7 * (math.Pi / (30 * 60))},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := minutesInRadians(c.time)
            if got != c.angle {
                t.Fatalf("Wanted %v radians, but got %v", c.angle, got)
            }
        })
    }
}
How much is that tiny little bit? Well...
Sixty seconds in a minute
thirty minutes in a half turn of the circle (math.Pi radians)
so 30 * 60 seconds in a half turn.
So if the time is 7 seconds past the hour ...
... we're expecting to see the minute hand at 7 * (math.Pi / (30 * 60))
radians past the 12.
Try to run the test
--- FAIL: TestMinutesInRadians (0.00s)
    --- FAIL: TestMinutesInRadians/00:00:07 (0.00s)
        clockface_test.go:62: Wanted 0.012217304763960306 radians, but got 3.141592653589793
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v8/clockface    0.009s
Write enough code to make it pass
In the immortal words of Jennifer Aniston: Here comes the science bit​
func minutesInRadians(t time.Time) float64 {
    return (secondsInRadians(t) / 60) +
        (math.Pi / (30 / float64(t.Minute())))
}
Rather than working out how far to push the minute hand around the clockface for every second from scratch, here we can just leverage the secondsInRadians function. For every second the minute hand will move 1/60th of the angle the second hand moves.
secondsInRadians(t) / 60
Then we just add on the movement for the minutes - similar to the movement of the second hand.
math.Pi / (30 / float64(t.Minute()))
And...
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v8/clockface    0.007s
Nice and easy.
Repeat for new requirements
Should I add more cases to the minutesInRadians test? At the moment there are only two. How many cases do I need before I move on to the testing the minuteHandPoint function?
One of my favourite TDD quotes, often attributed to Kent Beck, is
Write tests until fear is transformed into boredom.
And, frankly, I'm bored of testing that function. I'm confident I know how it works. So it's on to the next one.
Write the test first
func TestMinuteHandPoint(t *testing.T) {
    cases := []struct {
        time  time.Time
        point Point
    }{
        {simpleTime(0, 30, 0), Point{0, -1}},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := minuteHandPoint(c.time)
            if !roughlyEqualPoint(got, c.point) {
                t.Fatalf("Wanted %v Point, but got %v", c.point, got)
            }
        })
    }
}
Try to run the test
# github.com/gypsydave5/learn-go-with-tests/math/v9/clockface [github.com/gypsydave5/learn-go-with-tests/math/v9/clockface.test]
./clockface_test.go:79:11: undefined: minuteHandPoint
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v9/clockface [build failed]
Write the minimal amount of code for the test to run and check the failing test output
func minuteHandPoint(t time.Time) Point {
    return Point{}
}
--- FAIL: TestMinuteHandPoint (0.00s)
    --- FAIL: TestMinuteHandPoint/00:30:00 (0.00s)
        clockface_test.go:80: Wanted {0 -1} Point, but got {0 0}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.007s
Write enough code to make it pass
func minuteHandPoint(t time.Time) Point {
    return Point{0, -1}
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.007s
Repeat for new requirements
And now for some actual work
func TestMinuteHandPoint(t *testing.T) {
    cases := []struct {
        time  time.Time
        point Point
    }{
        {simpleTime(0, 30, 0), Point{0, -1}},
        {simpleTime(0, 45, 0), Point{-1, 0}},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := minuteHandPoint(c.time)
            if !roughlyEqualPoint(got, c.point) {
                t.Fatalf("Wanted %v Point, but got %v", c.point, got)
            }
        })
    }
}
--- FAIL: TestMinuteHandPoint (0.00s)
    --- FAIL: TestMinuteHandPoint/00:45:00 (0.00s)
        clockface_test.go:81: Wanted {-1 0} Point, but got {0 -1}
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.007s
Write enough code to make it pass
A quick copy and paste of the secondHandPoint function with some minor changes ought to do it...
func minuteHandPoint(t time.Time) Point {
    angle := minutesInRadians(t)
    x := math.Sin(angle)
    y := math.Cos(angle)
​
    return Point{x, y}
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.009s
Refactor
We've definitely got a bit of repetition in the minuteHandPoint and secondHandPoint - I know because we just copied and pasted one to make the other. Let's DRY it out with a function.
func angleToPoint(angle float64) Point {
    x := math.Sin(angle)
    y := math.Cos(angle)
​
    return Point{x, y}
}
and we can rewrite minuteHandPoint and secondHandPoint as one liners:
func minuteHandPoint(t time.Time) Point {
    return angleToPoint(minutesInRadians(t))
}
func secondHandPoint(t time.Time) Point {
    return angleToPoint(secondsInRadians(t))
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.007s
Now we can uncomment the acceptance test and get to work drawing the minute hand
Write enough code to make it pass
Another quick copy-and-paste with some minor adjustments
func minuteHand(w io.Writer, t time.Time) {
    p := minuteHandPoint(t)
    p = Point{p.X * minuteHandLength, p.Y * minuteHandLength}
    p = Point{p.X, -p.Y}
    p = Point{p.X + clockCentreX, p.Y + clockCentreY}
    fmt.Fprintf(w, `<line x1="150" y1="150" x2="%.3f" y2="%.3f" style="fill:none;stroke:#000;stroke-width:3px;"/>`, p.X, p.Y)
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.006s
But the proof of the pudding is in the eating - if we now compile and run our clockface program, we should see something like
a clock with second and minute hands
Refactor
Let's remove the duplication from the secondHand and minuteHand functions, putting all of that scale, flip and translate logic all in one place.
func secondHand(w io.Writer, t time.Time) {
    p := makeHand(secondHandPoint(t), secondHandLength)
    fmt.Fprintf(w, `<line x1="150" y1="150" x2="%.3f" y2="%.3f" style="fill:none;stroke:#f00;stroke-width:3px;"/>`, p.X, p.Y)
}
​
func minuteHand(w io.Writer, t time.Time) {
    p := makeHand(minuteHandPoint(t), minuteHandLength)
    fmt.Fprintf(w, `<line x1="150" y1="150" x2="%.3f" y2="%.3f" style="fill:none;stroke:#000;stroke-width:3px;"/>`, p.X, p.Y)
}
​
func makeHand(p Point, length float64) Point {
    p = Point{p.X * length, p.Y * length}
    p = Point{p.X, -p.Y}
    return Point{p.X + clockCentreX, p.Y + clockCentreY}
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v9/clockface    0.007s
There... now it's just the hour hand to do!
Write the test first
func TestSVGWriterHourHand(t *testing.T) {
    cases := []struct {
        time time.Time
        line Line
    }{
        {
            simpleTime(6, 0, 0),
            Line{150, 150, 150, 200},
        },
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            b := bytes.Buffer{}
            clockface.SVGWriter(&b, c.time)
​
            svg := SVG{}
            xml.Unmarshal(b.Bytes(), &svg)
​
            if !containsLine(c.line, svg.Line) {
                t.Errorf("Expected to find the hour hand line %+v, in the SVG lines %+v", c.line, svg.Line)
            }
        })
    }
}
Try to run the test
--- FAIL: TestSVGWriterHourHand (0.00s)
    --- FAIL: TestSVGWriterHourHand/06:00:00 (0.00s)
        clockface_acceptance_test.go:113: Expected to find the hour hand line {X1:150 Y1:150 X2:150 Y2:200}, in the SVG lines [{X1:150 Y1:150 X2:150 Y2:60} {X1:150 Y1:150 X2:150 Y2:70}]
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v10/clockface    0.013s
Again, let's comment this one out until we've got the some coverage with the lower level tests:
Write the test first
func TestHoursInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(6, 0, 0), math.Pi},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := hoursInRadians(c.time)
            if got != c.angle {
                t.Fatalf("Wanted %v radians, but got %v", c.angle, got)
            }
        })
    }
}
Try to run the test
# github.com/gypsydave5/learn-go-with-tests/math/v10/clockface [github.com/gypsydave5/learn-go-with-tests/math/v10/clockface.test]
./clockface_test.go:97:11: undefined: hoursInRadians
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v10/clockface [build failed]
Write the minimal amount of code for the test to run and check the failing test output
func hoursInRadians(t time.Time) float64 {
    return math.Pi
}
PASS
ok      github.com/gypsydave5/learn-go-with-tests/math/v10/clockface    0.007s
Repeat for new requirements
func TestHoursInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(6, 0, 0), math.Pi},
        {simpleTime(0, 0, 0), 0},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := hoursInRadians(c.time)
            if got != c.angle {
                t.Fatalf("Wanted %v radians, but got %v", c.angle, got)
            }
        })
    }
}
Try to run the test
--- FAIL: TestHoursInRadians (0.00s)
    --- FAIL: TestHoursInRadians/00:00:00 (0.00s)
        clockface_test.go:100: Wanted 0 radians, but got 3.141592653589793
FAIL
exit status 1
FAIL    github.com/gypsydave5/learn-go-with-tests/math/v10/clockface    0.007s
Write enough code to make it pass
func hoursInRadians(t time.Time) float64 {
    return (math.Pi / (6 / float64(t.Hour())))
}
Repeat for new requirements
func TestHoursInRadians(t *testing.T) {
    cases := []struct {
        time  time.Time
        angle float64
    }{
        {simpleTime(6, 0, 0), math.Pi},
        {simpleTime(0, 0, 0), 0},
        {simpleTime(21, 0, 0), math.Pi * 1.5},
    }
​
    for _, c := range cases {
        t.Run(testName(c.time), func(t *testing.T) {
            got := hoursInRadians(c.time)
            if got != c.angle {
                t.Fatalf(</