Learn some more vector math, and see how we can use it to implement gameplay movement.

Posted

Note: if following along with this chapter, it will be useful to have a relatively thorough implementation of the `Vector3`

struct. An example is available here:

```
#pragma once
struct Vector3 {
float x;
float y;
float z;
float GetLength() const {
return std::sqrt(x * x + y * y + z * z);
}
float GetDistance(const Vector3& Other) const {
using namespace std;
return sqrt(
pow(x - Other.x, 2) +
pow(y - Other.y, 2) +
pow(z - Other.z, 2)
);
}
Vector3 operator+ (const Vector3& Other) const {
return Vector3 {
x + Other.x,
y + Other.y,
z + Other.z
};
}
Vector3 operator- (const Vector3& Other) const {
return Vector3 {
x - Other.x,
y - Other.y,
z - Other.z
};
}
Vector3 operator* (float Multiplier) const {
return Vector3 {
x * Multiplier,
y * Multiplier,
z * Multiplier
};
}
Vector3 operator/ (float Divisor) const {
return Vector3 {
x / Divisor,
y / Divisor,
z / Divisor
};
}
Vector3 operator+= (const Vector3& Other) {
x += Other.x;
y += Other.y;
z += Other.z;
return *this;
};
Vector3 operator-= (const Vector3& Other) {
x -= Other.x;
y -= Other.y;
z -= Other.z;
return *this;
};
Vector3 operator*= (float Multiplier) {
x *= Multiplier;
y *= Multiplier;
z *= Multiplier;
return *this;
};
Vector3 operator/= (float Divisor) {
x /= Divisor;
y /= Divisor;
z /= Divisor;
return *this;
};
Vector3 operator- () const {
return Vector3 {-x, -y, -z };
};
};
```

This struct is similar to what we've been creating in previous chapters. The only difference is that we've made sure it is *const correct*. All function arguments that are references are marked as `const`

, and class functions that do not modify the underlying object are also marked as `const`

.

`const`

but others are notA common point of confusion here is why an operator like `--`

cannot be marked as `const`

but the unary `-`

can (and should) be marked as `const`

This is simply a question of what the operators actually do to the object they are called on. The `--`

operator modifies the object it is called on; the `-`

operator does not.

An expression like `--MyVector`

modifies `MyVector`

. This operator changes the values of `x`

, `y`

and `z`

variables of `MyVector`

, therefore it cannot be marked as `const`

.

An expression like `-MyVector`

does *not* modify `MyVector`

. The operator creates and returns a *new* vector object, with different `x`

, `y`

and `z`

values. Because it does not modify the original vector on which it is used, this operator should be marked as `const`

.

Vectors have more uses than storing where our objects currently are - they can also be used to calculate and store movements.

By storing a movement as vector, we can determine and visualise both the direction and distance of that movement.

To calculate an objects position after applying this movement, we add the movement vector to its current position. For example, imagine our character was initially at position `{ 5, 1 }`

and a movement vector of `{ 1, 3 }`

was applied.

The character's new position will be the result of adding those two vectors: `{ 5+1, 1+3 }`

which is `{ 6, 4 }`

We can add a function to our class to make that happen:

```
#include "Vector3.h"
class Character {
public:
Vector3 WorldPosition { 3.0, 1.0, 0.0 };
void Move(const Vector3& Movement) {
WorldPosition += Movement;
}
};
Character MyCharacter;
Vector3 MovementVector { 3.0, 2.0, 0.0 };
MyCharacter.Move(MovementVector);
```

Typically, we don't want our objects to be able to immediately move to a position. Rather than teleporting, we often want characters to move towards a location over time. For example, it might take a character 3 turns, or 100 frames to reach their destination.

One way we could implement this is through adding a `MovementSpeed`

variable to our `Character`

class. Then, we can implement a movement function such that it respects these limits.

To set this up, we need to take three steps:

- Create a
**normalised vector**from the movement vector - Multiply the normalised vector by the character's
`MovementSpeed`

to create a new vector - Add this new vector to the character's
`WorldPosition`

We previously saw how we could calculate the length (or magnitude) of a vector using the Pythagorean theorem.

$\sqrt{x^2 + y^2 + z^2}$

We made this available as the `GetLength`

function within our struct.

Normalising a vector means finding another vector that has *the same direction*, but a length of `1`

.

To do this, we calculate the length of the movement vector, and then divide the vector by this number.

Lets imagine we have the vector `{ 234.0, 105.0, 279.0 }`

. Applying the pythagorean theorem, we'd find the length of this vector turns out to be `379.0`

.

So, to normalise this vector, we divide it by `379.0`

. This yields approximately `{ 0.617, 0.277, 0.736 }`

. If we apply the pythagorean theorem to calculate the length of this new vector, we'd find it to be approximately `1.0`

, as planned.

That means `{ 0.617, 0.277, 0.736 }`

is the normalised form of `{ 234.0, 105.0, 279.0}`

Lets give our `Vector3`

struct the ability to normalise `Vector3`

objects:

```
struct Vector3 {
float x;
float y;
float z;
Vector3 Normalise() const {
return *this / GetLength();
}
// Rest of struct
}
```

If a vector has a length of 1, it is sometimes called a *unit vector*.

For example, `{ 1.0, 0.0, 0.0 }`

and `{ 0.0, 1.0, 0.0 }`

are unit vectors. The vector we calculated in this section - `{ 0.617, 0.277, 0.736 }`

- is also a unit vector.

Lets update our `Character`

objects with a new `MovementSpeed`

variable. We can then calculate our final movement vector by multiplying the normalised input vector with the character's maxiumum `MovementSpeed`

.

This ensures the character moves in the same direction requested by the input vector, but doesn't move further in that direction than their movement speed allows:

```
class Character {
public:
Vector3 WorldPosition { 2.0, 6.0, 0.0 };
float MovementSpeed { 10.0 };
void Move(const Vector3& Movement) {
Vector3 MovementVector =
Movement.Normalise() * MovementSpeed;
}
};
```

Finally, our last step is to add this new movement vector to the character's `WorldPosition`

, thereby making the character move:

```
class Character {
public:
Vector3 WorldPosition { 2.0, 6.0, 0.0 };
float MovementSpeed { 10.0 };
void Move(const Vector3& Movement) {
Vector3 MovementVector =
Movement.Normalise() * MovementSpeed;
WorldPosition += MovementVector;
}
};
```

In this code, the character will always move the maximum distance permitted by their `MovementSpeed`

. This means there is a bug when the character is told to move a distance shorter than their movement speed: they will overshoot the target.

We can check for this scenario, and fix this with an `if`

statement. If the length of the input vector is less than or equal to the character's maximum `MovementSpeed`

, we can just move the full length of that vector immediately:

```
void Move(const Vector3& Movement) {
// Can the character reach their destination this frame?
if (Movement.GetLength() <= MovementSpeed) {
WorldPosition += Movement;
return;
}
Vector3 MovementVector =
Movement.Normalise() * MovementSpeed;
WorldPosition += MovementVector;
}
```

Having our `Move`

function require a movement vector be provided is not especially friendly to anyone using our code.

Supporting an instruction like "move towards this other object" is going to be more useful that "move along this vector".

If we have two points in space, represented by the vectors `A`

and `B`

. The vector that moves from `A`

to `B`

is the result of subtracting `A`

from `B`

, which is `B - A`

Lets use this to add the more useful `MoveTowards`

function to our `Character`

:

```
class Character {
public:
Vector3 WorldPosition { 2.0, 6.0, 0.0 };
float MovementSpeed { 10.0 };
void Move(const Vector3& Movement) {
if (Movement.GetLength() < MovementSpeed) {
WorldPosition += Movement;
return;
}
Vector3 MovementVector =
Movement.Normalise() * MovementSpeed;
WorldPosition += MovementVector;
}
void MoveTowards(const Character& Target) {
Move(Target.WorldPosition - WorldPosition);
}
};
```

We might want to make an evasive archer character that prefers to fight at range, moving away from her targets.

To make a character move away from a target, we can simply invert the order of operations:

```
void MoveAway(const Character& Target) {
Move(WorldPosition - Target.WorldPosition);
}
```

This works because of a useful property of vectors: `SomeVector`

and `-SomeVector`

have the same length, but point in opposite directions.

With movement mechanics now available to use, we can create much more dynamic combat simulations.

Currently, every time we run our program, the results will be the same. In the next section, we will introduce a way to create randomness.

Posted

This lesson is part of the course:### Intro to C++ Programming

Become a software engineer with C++. Starting from the basics, we guide you step by step along the way