Angles

This article is a continuation of Mouse Movement, with a focus on how trigonometry works in Minecraft.

The Player's yaw is a float that keeps track of the player's horizontal rotation.

The Player's facing is the restriction of the yaw to [-180, 180].

A significant angle is an integer (from ${\textstyle 0}$ to ${\textstyle 2^{16}-1}$).

Significant Angles

Minecraft relies on significant angles for its trigonometry, which means the Player's facing has to be converted one such angle.

This conversion induces imprecision, which means two close facings can be attributed the same angle (a significant angle spans across ~Failed to parse (syntax error): {\displaystyle 0.0055°} ).

Sin() and Cos() source code (from MathHelper):

private static final float[] SIN_TABLE = new float[65536];


public static float sin(float value) //in radians
{
    return SIN_TABLE[(int)(value * 10430.378F) & 65535];
}

public static float cos(float value) //in radians
{
    return SIN_TABLE[(int)(value * 10430.378F + 16384.0F) & 65535];
}


static
{
    for (int i = 0; i < 65536; ++i)
    {
        SIN_TABLE[i] = (float)Math.sin((double)i * Math.PI * 2.0D / 65536.0D);
    }
}

Note: & 65535 gives the (positive) remainder of a division by 65536 ( $2^{16}$)

Half Angles

Ideally, (int)(value * 10430.378F) and (int)(value * 10430.378F + 16384.0F) would be 16384 units apart (90°). But because of float imprecision, some values could be further by 1 unit, causing a slight shift from the intended calculation.

Half angles are such values, and can found "between" consecutive angles (hence the name).

Two consecutive angles (30237 and 30238) and a positive half angle in between.

Half angles don't have much use outside of Tool-Assisted Parkour: their effect on jump distance is negligible, and they are hardly usable with mouse movement.

• Positive half-angles increase the Player's speed. They can be found (rarely) in the North-West quadrant (90° to 180°)
• Negative half-angles decrease the Player's speed. They can be found (abundantly) in the South-East quadrant (-90° to 0°), as well as in the South-West quadrant (0° to 90°) in rare amounts.

Each half-angle has an associated Multiplier that represents its effectiveness.

It corresponds to the norm of the unit vector obtained from cos(F) and sin(F). Mathematically, it should always be equal to 1, but that doesn't hold true in the context of half-angles.

${\textstyle \left\|{\vec {v}}\right\|={\sqrt {\cos(F)^{2}+\sin(F)^{2}}}}$

When multiplied with a given jump distance, it gives an upper bound for the improved jump distance with its corresponding half-angle.

List of all positive Half Angles

Graph showing the relative effect of Half Angles. A multiplier above 1 indicates a positive half angle.

Note: Fast Math is an Optifine feature that reduces the number of angles to 4096 ( $2^{12}$).

With Fast Math, half angles are fewer in number, but are up to 16x more effective. For example, it makes a no-sprint 3b jump possible with only flat momentum.