Naval Battle

Summary:

1. The 3D engine
   1. Global idea
2. The physics engine
   1. Rigid body dynamics
   2. Water simulation
   3. Water and rigid body contacts
   4. Damaging rigid bodies
3. Gameplay spécifications
   1. Game rules
   2. Techniques

1. The 3D engine

The 3D engine is written in C++, using openGL. The main goal of this part is to understand how openGL work. In fact, the course we got hasn't got enough in depth for me.

1.1 Global idea

2. The physics engine

The physics engine is really important for the gameplay. The goal is to have something complex, with plenty of possibilities, but intuitive. In this chapter, we will describe how it works.

2.1 Rigid Body Dynamics

A Rigid Body is a body without deformations.

2.1.1 Informations needed to describe a rigid body

It has a position (a 3D vector), describing the position of his center of mass. To describe his movement, it has a speed (a 3D vector) and a torque for each axis (a 3D vector). It has a mass (a scalar), and a mass moment of inertia (we will call it inertia) (a 3x3 matrix)

2.1.2 Applying a force on a rigid body

We have a rigid body $A$ with a center of mass $c_A$. We want to put a force $f$ on the point $p_A$ , and we want to know what is his new acceleration, and his new rotational acceleration.
If $c_A - p_A$ and $f$ are aligned, we don't have any rotation. So applying the force is pretty straitforward.
The new $acceleration$ is $acceleration$ += $\frac f {mass}$
If $c_A - p_A$ and $f$ are not aligned, we need to add some rotational speed too.
$torque$ += $I^{-1} \cdot ((p_A - c_A) \times f)$
(" $\times$ " is for the cross product, and " $\cdot$ " is for the matrix-vector multiplication)

N.B.: $I^{-1}$ should be calculated only once, same for $\frac 1 {mass}$

2.1.3 Applying an impulse on a rigid body

It follows the same ideas. The unique difference is that we don't change the acceleration, but the speed.

2.1.4 Resolve collisions

This is one of the hardest part.
We have two rigid bodies colliding: A and B. We suppose they are colliding only in a point or a line. Their velocities before the contact are $v_A$ and $v_B$. $v$ is the relative velocity, i.e. $v_B - v_A$. the contact point is $p$. If they are colliding in a line, we will compute this twice, the first time for an extremity of the line, and the second time for the other extremity. $n$ is the normal of the contact. $e$ is the coefficiant of restitution (between 0 and 1). $r_A$ is equal to $p - c_A$, and $r_B$ is equal to $p - c_B$. Finally, let $R_A = (f - \frac {f \cdot r_A} {|r_A|^2} r_A)$.resize($|r_A|$). $R_A$ is the projection of $f$ on the orthogonal plane of $r_A$, with the size of $r_A$.

We want to know the velocities after the impact.

$$j = \frac {-(1+e) v \cdot n} {n \cdot n (\frac 1 {m_A} + \frac 1 {m_B}) + I_A^{-1} (R_A \cdot n)^2 + I_B^{-1} (R_B \cdot n)^2}$$

Then, we can change the velocities of the bodies using the impulses:
$v_A$ += $\frac j {m_A} n$
$t_A$ += $I_A ^{-1} (R_A \cdot j n)$
$v_B$ -= $\frac j {m_B} n$
$t_B$ -= $I_B ^{-1} (R_B \cdot j n)$
($t_A$ is the torque and $v_A$ is the linear velocity.)

For further reading: https://www.chrishecker.com/Rigid_Body_Dynamics.

2.1.5 Frictions

Only with that, the rigid bodies are pretty slippery. In fact, we need to add friction during the collisions.
The first question is, how do we stock this ? Normally, we use a friction table that describe the friction constants between two materials. Here, in order to simplify, we give some friction constants to each material. So, they have a staticFriction and a dynamicFriction, between 0 and 1.

2. Gameplay specifications

It is a turn-based strategy.

2.1 Game rules

2.1.1 Placements

There are 2 teams fighting. You control one of this two teams. Each team has 3 navigators and as many ships as they want (usually, there are 3 ships, one for each navigator). One of this ships has a gol, a little crystal.

2.1.2 Goal

There are multiple ways to win. One way is to destroy the gol of the other team, or let it touch the floor under the water. This method is less valuable, but easyer to do. The other way is to steel the gol, and place it in your base.

2.2 Techniques

Each navigator is a wizard. Each navigator has a certain amount of mana. Using a power cost some mana. The cost vary proportionnaly to the distance with the caster, and the strength of the power used.

2.2.1 Base techniques

Each navigator can use this techniques

Telekinesis

The navigator can apply a force to an object. Very useful to move the boat.

2.2.2 Alchemist

An alchemist can transform the elements.

2.2.3 Runist

A runist is someone who create and use runes.

2.2.4 Invoker

An invoker can create things.