# Angles: Difference between revisions

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This article is a continuation of [[Special:MyLanguage/Mouse Movement|Mouse Movement]], with a focus on how trigonometry works in Minecraft. |
This article is a continuation of [[Special:MyLanguage/Mouse Movement|Mouse Movement]], with a focus on how trigonometry works in Minecraft. |
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The player's '''yaw''' is a [https://en.wikipedia.org/wiki/Floating-point_arithmetic float] that keeps track of the player's horizontal rotation (in degrees). It is '''unbounded'''. |
The player's '''yaw''' is a [https://en.wikipedia.org/wiki/Floating-point_arithmetic float] that keeps track of the player's horizontal rotation (in degrees). It is '''unbounded'''. |
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The player's '''facing''' is the restriction of the yaw to [-180°, 180°], as shown in [[Special:MyLanguage/Debug Screen|F3]]. |
The player's '''facing''' is the restriction of the yaw to [-180°, 180°], as shown in [[Special:MyLanguage/Debug Screen|F3]]. |
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A '''significant angle''', or simply '''angle''' is an integer (from <math display="inline">0</math> to <math display="inline">2^{16}-1</math>). |
A '''significant angle''', or simply '''angle''' is an integer (from <math display="inline">0</math> to <math display="inline">2^{16}-1</math>). |
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− | == Significant Angles == |
+ | == Significant Angles == <!--T:5--> |

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Minecraft relies on significant angles for its trigonometry, which means the player's yaw has to be converted to an angle. |
Minecraft relies on significant angles for its trigonometry, which means the player's yaw has to be converted to an angle. |
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This conversion induces imprecision: a significant angle spans across ~<math>0.0055°</math>). |
This conversion induces imprecision: a significant angle spans across ~<math>0.0055°</math>). |
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− | '''Sin() and Cos() source code''' (from |
+ | '''Sin() and Cos() source code''' (from class MathHelper):<syntaxhighlight lang="java"> |

private static final float[] SIN_TABLE = new float[65536]; |
private static final float[] SIN_TABLE = new float[65536]; |
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public static float sin(float value) //in radians |
public static float sin(float value) //in radians |
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{ |
{ |
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} |
} |
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public static float cos(float value) //in radians |
public static float cos(float value) //in radians |
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{ |
{ |
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static |
static |
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{ |
{ |
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</syntaxhighlight>'''Note:''' ''<code>& 65535</code>'' gives the (positive) remainder of a division by 65536 ( <math>2^{16}</math>) |
</syntaxhighlight>'''Note:''' ''<code>& 65535</code>'' gives the (positive) remainder of a division by 65536 ( <math>2^{16}</math>) |
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To convert the player's yaw into radians (when calling sin and cos), the game uses two formulas:<syntaxhighlight lang="java"> |
To convert the player's yaw into radians (when calling sin and cos), the game uses two formulas:<syntaxhighlight lang="java"> |
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//formula used in general |
//formula used in general |
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f = this.rotationYaw * (float)Math.PI / 180.0F |
f = this.rotationYaw * (float)Math.PI / 180.0F |
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//formula used when adding sprintjump boost |
//formula used when adding sprintjump boost |
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f = this.rotationYaw * 0.017453292F |
f = this.rotationYaw * 0.017453292F |
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Both have the same intent, but for large values the result may be different. |
Both have the same intent, but for large values the result may be different. |
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In that case, sprintjumping moves the player slightly to the side, which may be useful for no-turn strats. |
In that case, sprintjumping moves the player slightly to the side, which may be useful for no-turn strats. |
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− | |||

⚫ | |||

− | |||

<code>(int)(value * 10430.378F)</code> and <code>(int)(value * 10430.378F + 16384.0F)</code> should be 16384 units apart (90°), but because of floating point imprecision, some values could end up 1 or more units further, causing a slight shift from the intended calculation. |
<code>(int)(value * 10430.378F)</code> and <code>(int)(value * 10430.378F + 16384.0F)</code> should be 16384 units apart (90°), but because of floating point imprecision, some values could end up 1 or more units further, causing a slight shift from the intended calculation. |
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[[File:Half angle visualized.png|thumb|582x582px|Two consecutive angles (30237 and 30238) and a positive half angle in between.]]'''Half angles''' are such values, and can be found "between" consecutive angles (hence the name). |
[[File:Half angle visualized.png|thumb|582x582px|Two consecutive angles (30237 and 30238) and a positive half angle in between.]]'''Half angles''' are such values, and can be found "between" consecutive angles (hence the name). |
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The existence of half angles is entirely due to the fact the term "+ 16384" is inside the parentheses. |
The existence of half angles is entirely due to the fact the term "+ 16384" is inside the parentheses. |
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Half angles don't have much use outside of Tool-Assisted Parkour: their effect on jump distance is small, and they are hardly usable with real-time mouse movement. |
Half angles don't have much use outside of Tool-Assisted Parkour: their effect on jump distance is small, and they are hardly usable with real-time mouse movement. |
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Each half-angle has an associated '''Multiplier''' that represents its effectiveness. |
Each half-angle has an associated '''Multiplier''' that represents its effectiveness. |
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It corresponds to the norm of the unit vector obtained from cos(f) and sin(f). |
It corresponds to the norm of the unit vector obtained from cos(f) and sin(f). |
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Mathematically, it should always be equal to 1, but that doesn't hold true in this context. |
Mathematically, it should always be equal to 1, but that doesn't hold true in this context. |
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* "Increasing" half angles have a norm greater than 1: they increase movement speed. |
* "Increasing" half angles have a norm greater than 1: they increase movement speed. |
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* "Decreasing" half angles have a norm lesser than 1: they decrease movement speed. |
* "Decreasing" half angles have a norm lesser than 1: they decrease movement speed. |
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<math display="inline">\left \|\vec{v}\right \|=\sqrt{\cos(f)^{2}+\sin(f)^{2}}</math> |
<math display="inline">\left \|\vec{v}\right \|=\sqrt{\cos(f)^{2}+\sin(f)^{2}}</math> |
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When multiplied with a given jump distance, it gives an upper bound for the improved jump distance with its corresponding half-angle. |
When multiplied with a given jump distance, it gives an upper bound for the improved jump distance with its corresponding half-angle. |
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[[Special:MyLanguage/Positive Half Angles|List of useful Half Angles]] |
[[Special:MyLanguage/Positive Half Angles|List of useful Half Angles]] |
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− | == Large Half Angles == |
+ | == Large Half Angles == <!--T:27--> |

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On July 27th 2021, kemytz discovered that [https://youtu.be/eMDajBi_QW0 certain yaws grant more speed] when using Optifine Fast Math (a feature which reduces the number of significant angles down to 4096). In fact, this is not specific to fast math, and vanilla "large half angles" were discovered soon after. |
On July 27th 2021, kemytz discovered that [https://youtu.be/eMDajBi_QW0 certain yaws grant more speed] when using Optifine Fast Math (a feature which reduces the number of significant angles down to 4096). In fact, this is not specific to fast math, and vanilla "large half angles" were discovered soon after. |
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Large half angles are more effective due to the low amount of precision floats can work with at that range: instead of being shifted by a single unit, angles can be shifted by up to 64 units. |
Large half angles are more effective due to the low amount of precision floats can work with at that range: instead of being shifted by a single unit, angles can be shifted by up to 64 units. |
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+ | <!--T:30--> |
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For example, the yaw 5898195° gives a multiplier of 1.003, which is huge compared to small half angles (135.0055° gives a multiplier of 1.00005). This half angle is the most effective that exists, and makes jumps such as [https://youtu.be/bQfl2Lleafw 2.125bm 4+0.5] possible. |
For example, the yaw 5898195° gives a multiplier of 1.003, which is huge compared to small half angles (135.0055° gives a multiplier of 1.00005). This half angle is the most effective that exists, and makes jumps such as [https://youtu.be/bQfl2Lleafw 2.125bm 4+0.5] possible. |
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Reaching such high values may require using macros or mods, as turning manually would take too long. |
Reaching such high values may require using macros or mods, as turning manually would take too long. |
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With Fast Math, half angles are further amplified and the player is able to move while their yaw is less than <math>360 \times 2^{19}</math>, compared to <math>360 \times 2^{15}</math> in vanilla. |
With Fast Math, half angles are further amplified and the player is able to move while their yaw is less than <math>360 \times 2^{19}</math>, compared to <math>360 \times 2^{15}</math> in vanilla. |
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For example, the yaw 121000000° gives a multiplier of 1.09, which is so ludicrous it makes [https://youtu.be/Juz9bDZa5rc flat momentum 5b possible]. This mechanic should not be considered official as it relies on a mod, and using this exploit is likely not authorized on servers (even if Optifine is). |
For example, the yaw 121000000° gives a multiplier of 1.09, which is so ludicrous it makes [https://youtu.be/Juz9bDZa5rc flat momentum 5b possible]. This mechanic should not be considered official as it relies on a mod, and using this exploit is likely not authorized on servers (even if Optifine is). |
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+ | <!--T:34--> |
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New versions of Optifine changed the effect of fast math, which eliminated a lot of FM half angles (since version U L5, released in december 2019). |
New versions of Optifine changed the effect of fast math, which eliminated a lot of FM half angles (since version U L5, released in december 2019). |
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− | == Characterization == |
+ | == Characterization == <!--T:35--> |

+ | <!--T:36--> |
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Decreasing half angles are abundant between -90° and 0°, because negative floats are truncated up while positive floats are truncated down. |
Decreasing half angles are abundant between -90° and 0°, because negative floats are truncated up while positive floats are truncated down. |
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+ | <!--T:37--> |
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Decreasing half angles exist rarely between 0° and 90°. |
Decreasing half angles exist rarely between 0° and 90°. |
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+ | <!--T:38--> |
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Increasing half angles exist rarely between 90° and 180°. |
Increasing half angles exist rarely between 90° and 180°. |
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Large half angles are of the form <math>360 \times 2^n - \theta</math>, with <math>\theta \in [0,90]</math> and <math>n \in \{0, ... ,14\}</math>. They are all increasing until <math>n \geq 8</math>, at which point they may either be increasing or decreasing. |
Large half angles are of the form <math>360 \times 2^n - \theta</math>, with <math>\theta \in [0,90]</math> and <math>n \in \{0, ... ,14\}</math>. They are all increasing until <math>n \geq 8</math>, at which point they may either be increasing or decreasing. |
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</translate> |
</translate> |

## Latest revision as of 18:51, 3 September 2022

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 (in degrees). It is **unbounded**.

The player's **facing** is the restriction of the yaw to [-180°, 180°], as shown in F3.

A **significant angle**, or simply **angle** is an integer (from to ).

## Significant Angles

Minecraft relies on significant angles for its trigonometry, which means the player's yaw has to be converted to an angle.

This conversion induces imprecision: a significant angle spans across ~).

**Sin() and Cos() source code**(from class 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:**

*gives the (positive) remainder of a division by 65536 ( ) To convert the player's yaw into radians (when calling sin and cos), the game uses two formulas:*

`& 65535`

```
//formula used in general
f = this.rotationYaw * (float)Math.PI / 180.0F
//formula used when adding sprintjump boost
f = this.rotationYaw * 0.017453292F
```

Both have the same intent, but for large values the result may be different.

In that case, sprintjumping moves the player slightly to the side, which may be useful for no-turn strats.

## Half Angles

`(int)(value * 10430.378F)`

and `(int)(value * 10430.378F + 16384.0F)`

should be 16384 units apart (90°), but because of floating point imprecision, some values could end up 1 or more units further, causing a slight shift from the intended calculation.

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

The existence of half angles is entirely due to the fact the term "+ 16384" is inside the parentheses.

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

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 this context.

- "Increasing" half angles have a norm greater than 1: they increase movement speed.
- "Decreasing" half angles have a norm lesser than 1: they decrease movement speed.

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

## Large Half Angles

On July 27th 2021, kemytz discovered that certain yaws grant more speed when using Optifine Fast Math (a feature which reduces the number of significant angles down to 4096). In fact, this is not specific to fast math, and vanilla "large half angles" were discovered soon after.

Large half angles are more effective due to the low amount of precision floats can work with at that range: instead of being shifted by a single unit, angles can be shifted by up to 64 units.

For example, the yaw 5898195° gives a multiplier of 1.003, which is huge compared to small half angles (135.0055° gives a multiplier of 1.00005). This half angle is the most effective that exists, and makes jumps such as 2.125bm 4+0.5 possible.

Reaching such high values may require using macros or mods, as turning manually would take too long.

With Fast Math, half angles are further amplified and the player is able to move while their yaw is less than , compared to in vanilla.

For example, the yaw 121000000° gives a multiplier of 1.09, which is so ludicrous it makes flat momentum 5b possible. This mechanic should not be considered official as it relies on a mod, and using this exploit is likely not authorized on servers (even if Optifine is).

New versions of Optifine changed the effect of fast math, which eliminated a lot of FM half angles (since version U L5, released in december 2019).

## Characterization

Decreasing half angles are abundant between -90° and 0°, because negative floats are truncated up while positive floats are truncated down.

Decreasing half angles exist rarely between 0° and 90°.

Increasing half angles exist rarely between 90° and 180°.

Large half angles are of the form , with and . They are all increasing until , at which point they may either be increasing or decreasing.