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<h3 class="sectionHead"><span class="titlemark">1 </span> <a
id="x1-10001"></a>Introduction to 3D Math</h3>
<!--l. 27--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">1.1 </span> <a
id="x1-20001.1"></a>Introduction</h4>
<!--l. 29--><p class="noindent" >There are many approaches to understanding the type of 3D math used in video
games, modelling, ray-tracing, etc. The usual is through vector algebra, matrices, and
linear transformations and, while they are not completely necesary to understand
most of the aspects of 3D game programming (from the theorical point of view), they
provide a common language to communicate with other programmers or
engineers.
<!--l. 36--><p class="indent" > This tutorial will focus on explaining all the basic concepts needed for a
programmer to understand how to develop 3D games without getting too deep into
algebra. Instead of a math-oriented language, code examples will be given instead
when possible. The reason for this is that. while programmers may have
different backgrounds or experience (be it scientific, engineering or self taught),
code is the most familiar language and the lowest common denominator for
understanding.
<!--l. 45--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">1.2 </span> <a
id="x1-30001.2"></a>Vectors</h4>
<!--l. 48--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.1 </span> <a
id="x1-40001.2.1"></a>Brief Introduction</h5>
<!--l. 50--><p class="noindent" >When writing 2D games, interfaces and other applications, the typical convention is
to define coordinates as an <span
class="ecti-1000">x,y </span>pair, <span
class="ecti-1000">x </span>representing the horizontal offset and <span
class="ecti-1000">y </span>the
vertical one. In most cases, the unit for both is <span
class="ecti-1000">pixels</span>. This makes sense given the
screen is just a rectangle in two dimensions.
<!--l. 56--><p class="indent" > An <span
class="ecti-1000">x,y </span>pair can be used for two purposes. It can be an absolute position (screen
cordinate in the previous case), or a relative direction, if we trace an arrow from the
origin (0,0 coordinates) to it’s position.
<div class="center"
>
<!--l. 60--><p class="noindent" >
<div class="tabular">
<table id="TBL-1" class="tabular"
cellspacing="0" cellpadding="0"
><colgroup id="TBL-1-1g"><col
id="TBL-1-1"><col
id="TBL-1-2"><col
id="TBL-1-3"></colgroup><tr
style="vertical-align:baseline;" id="TBL-1-1-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-1"
class="td11"><img
src="tutorial0x.png" alt="PIC" class="graphics" width="100.375pt" height="100.375pt" ><!--tex4ht:graphics
name="tutorial0x.png" src="0_home_red_coding_godot_doc_math_position.eps"
--></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-2"
class="td11"></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-3"
class="td11"><img
src="tutorial1x.png" alt="PIC" class="graphics" width="100.375pt" height="100.375pt" ><!--tex4ht:graphics
name="tutorial1x.png" src="1_home_red_coding_godot_doc_math_direction.eps"
--></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-2-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-1"
class="td11"> <span
class="ecti-0700">Position </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-2"
class="td11"></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-3"
class="td11"> <span
class="ecti-0700">Direction </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-3-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-3-1"
class="td11"> </td>
</tr></table></div>
</div>
<!--l. 67--><p class="indent" > When used as a direction, this pair is called a <span
class="ecti-1000">vector</span>, and two properties can be
observed: The first is the <span
class="ecti-1000">magnitude </span>or <span
class="ecti-1000">length </span>, and the second is the direction. In
two dimensions, direction can be an angle. The <span
class="ecti-1000">magnitude </span>or <span
class="ecti-1000">length </span>can be computed
by simply using Pithagoras theorem:
<div class="center"
>
<!--l. 73--><p class="noindent" >
<div class="tabular"> <table id="TBL-2" class="tabular"
cellspacing="0" cellpadding="0"
><colgroup id="TBL-2-1g"><col
id="TBL-2-1"><col
id="TBL-2-2"></colgroup><tr
style="vertical-align:baseline;" id="TBL-2-1-"><td style="white-space:nowrap; text-align:center;" id="TBL-2-1-1"
class="td11"><img
src="tutorial2x.png" alt="∘x2-+-y2-" class="sqrt" ></td><td style="white-space:nowrap; text-align:center;" id="TBL-2-1-2"
class="td11"><img
src="tutorial3x.png" alt="∘x2-+-y2 +-z2" class="sqrt" ></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-2-"><td style="white-space:nowrap; text-align:center;" id="TBL-2-2-1"
class="td11"> <span
class="ecti-0700">2D </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-2-2-2"
class="td11"> <span
class="ecti-0700">3D </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-3-"><td style="white-space:nowrap; text-align:center;" id="TBL-2-3-1"
class="td11"> </td>
</tr></table></div>
</div>
<!--l. 80--><p class="indent" > The direction can be an arbitrary angle from either the <span
class="ecti-1000">x </span>or <span
class="ecti-1000">y </span>axis, and could be
computed by using trigonometry, or just using the usual <span
class="ecti-1000">atan2 </span>function present in
most math libraries. However, when dealing with 3D, the direction can’t be described
as an angle. To separate magnitude and direction, 3D uses the concept of <span
class="ecti-1000">normal</span>
<span
class="ecti-1000">vectors.</span>
<!--l. 88--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.2 </span> <a
id="x1-50001.2.2"></a>Implementation</h5>
<!--l. 90--><p class="noindent" >Vectors are implemented in Godot Engine as a class named <span
class="ecti-1000">Vector3 </span>for 3D, and as
both <span
class="ecti-1000">Vector2</span>, <span
class="ecti-1000">Point2 </span>or <span
class="ecti-1000">Size2 </span>in 2D (they are all aliases). They are used for any
purpose where a pair of 2D or 3D values (described as <span
class="ecti-1000">x,y </span>or <span
class="ecti-1000">x,y,z) </span>is needed. This is
somewhat a standard in most libraries or engines. In the script API, they can be
instanced like this:
<!--l. 98-->
<div class="lstlisting"><span class="label"><a
id="x1-5001r1"></a></span>a = Vector3() <br /><span class="label"><a
id="x1-5002r2"></a></span>a = Vector2( 2.0, 3.4 )
</div>
<!--l. 104--><p class="indent" > Vectors also support the common operators <span
class="ecti-1000">+, -, / and * </span>for addition,
substraction, multiplication and division.
<!--l. 108-->
<div class="lstlisting"><span class="label"><a
id="x1-5003r1"></a></span>a = Vector3(1,2,3) <br /><span class="label"><a
id="x1-5004r2"></a></span>b = Vector3(4,5,6) <br /><span class="label"><a
id="x1-5005r3"></a></span>c = Vector3() <br /><span class="label"><a
id="x1-5006r4"></a></span> <br /><span class="label"><a
id="x1-5007r5"></a></span>// writing <br /><span class="label"><a
id="x1-5008r6"></a></span> <br /><span class="label"><a
id="x1-5009r7"></a></span>c = a + b <br /><span class="label"><a
id="x1-5010r8"></a></span> <br /><span class="label"><a
id="x1-5011r9"></a></span>// is the same as writing <br /><span class="label"><a
id="x1-5012r10"></a></span> <br /><span class="label"><a
id="x1-5013r11"></a></span>c.x = a.x + b.x <br /><span class="label"><a
id="x1-5014r12"></a></span>c.y = a.y + b.y <br /><span class="label"><a
id="x1-5015r13"></a></span>c.z = a.z + b.z <br /><span class="label"><a
id="x1-5016r14"></a></span> <br /><span class="label"><a
id="x1-5017r15"></a></span>// both will result in a vector containing (5,7,9). <br /><span class="label"><a
id="x1-5018r16"></a></span>// the same happens for the rest of the operators.
</div>
<!--l. 128--><p class="indent" > Vectors also can perform a wide variety of built-in functions, their most common
usages will be explored next.
<!--l. 132--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.3 </span> <a
id="x1-60001.2.3"></a>Normal Vectors</h5>
<!--l. 134--><p class="noindent" >Two points ago, it was mentioned that 3D vectors can’t describe their direction as an
agle (as 2D vectors can). Because of this, <span
class="ecti-1000">normal vectors </span>become important for
separating a vector between <span
class="ecti-1000">direction </span>and <span
class="ecti-1000">magnitude.</span>
<!--l. 139--><p class="indent" > A <span
class="ecti-1000">normal vector </span>is a vector with a <span
class="ecti-1000">magnitude </span>of <span
class="ecti-1000">1. </span>This means, no matter where
the vector is pointing to, it’s length is always <span
class="ecti-1000">1</span>.
<div class="tabular">
<table id="TBL-3" class="tabular"
cellspacing="0" cellpadding="0"
><colgroup id="TBL-3-1g"><col
id="TBL-3-1"></colgroup><tr
style="vertical-align:baseline;" id="TBL-3-1-"><td style="white-space:nowrap; text-align:center;" id="TBL-3-1-1"
class="td11"><img
src="tutorial4x.png" alt="PIC" class="graphics" width="100.375pt" height="100.375pt" ><!--tex4ht:graphics
name="tutorial4x.png" src="2_home_red_coding_godot_doc_math_normals.eps"
--></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-3-2-"><td style="white-space:nowrap; text-align:center;" id="TBL-3-2-1"
class="td11"> <span
class="ecrm-0700">Normal vectors aroud the origin. </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-3-3-"><td style="white-space:nowrap; text-align:center;" id="TBL-3-3-1"
class="td11"> </td> </tr></table>
</div>
<!--l. 148--><p class="indent" > Normal vectors have endless uses in 3D graphics programming, so it’s
recommended to get familiar with them as much as possible.
<!--l. 152--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.4 </span> <a
id="x1-70001.2.4"></a>Normalization</h5>
<!--l. 154--><p class="noindent" >Normalization is the process through which normal vectors are obtained
from regular vectors. In other words, normalization is used to reduce the
<span
class="ecti-1000">magnitude </span>of any vector to <span
class="ecti-1000">1</span>. (except of course, unless the vector is (0,0,0)
).
<!--l. 159--><p class="indent" > To normalize a vector, it must be divided by its magnitude (which should be
greater than zero):
<!--l. 163-->
<div class="lstlisting"><span class="label"><a
id="x1-7001r1"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">custom</span><span
class="ecti-1000"> </span><span
class="ecti-1000">vector</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">created</span> <br /><span class="label"><a
id="x1-7002r2"></a></span>a = Vector3(4,5,6) <br /><span class="label"><a
id="x1-7003r3"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">’</span><span
class="ecti-1000">l</span><span
class="ecti-1000">’</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">single</span><span
class="ecti-1000"> </span><span
class="ecti-1000">real</span><span
class="ecti-1000"> </span><span
class="ecti-1000">number</span><span
class="ecti-1000"> </span><span
class="ecti-1000">(</span><span
class="ecti-1000">or</span><span
class="ecti-1000"> </span><span
class="ecti-1000">scalar</span><span
class="ecti-1000">)</span><span
class="ecti-1000"> </span><span
class="ecti-1000">containight</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">length</span> <br /><span class="label"><a
id="x1-7004r4"></a></span>l = Math.sqrt( a.x<span
class="cmsy-10">*</span>a.x + a.y<span
class="cmsy-10">*</span>a.y + a.z<span
class="cmsy-10">*</span>a.z ) <br /><span class="label"><a
id="x1-7005r5"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">vector</span><span
class="ecti-1000"> </span><span
class="ecti-1000">’</span><span
class="ecti-1000">a</span><span
class="ecti-1000">’</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">divided</span><span
class="ecti-1000"> </span><span
class="ecti-1000">by</span><span
class="ecti-1000"> </span><span
class="ecti-1000">its</span><span
class="ecti-1000"> </span><span
class="ecti-1000">length</span><span
class="ecti-1000">,</span><span
class="ecti-1000"> </span><span
class="ecti-1000">by</span><span
class="ecti-1000"> </span><span
class="ecti-1000">performing</span><span
class="ecti-1000"> </span><span
class="ecti-1000">scalar</span><span
class="ecti-1000"> </span><span
class="ecti-1000">divide</span> <br /><span class="label"><a
id="x1-7006r6"></a></span>a = a / l <br /><span class="label"><a
id="x1-7007r7"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">which</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">same</span><span
class="ecti-1000"> </span><span
class="ecti-1000">as</span> <br /><span class="label"><a
id="x1-7008r8"></a></span>a.x = a.x / l <br /><span class="label"><a
id="x1-7009r9"></a></span>a.y = a.y / l <br /><span class="label"><a
id="x1-7010r10"></a></span>a.z = a.z / l
</div>
<!--l. 177--><p class="indent" > Vector3 contains two built in functions for normalization:
<!--l. 180-->
<div class="lstlisting"><span class="label"><a
id="x1-7011r1"></a></span>a = Vector3(4,5,6) <br /><span class="label"><a
id="x1-7012r2"></a></span>a.normalize() <span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">in</span><span
class="cmsy-10">-</span><span
class="ecti-1000">place</span><span
class="ecti-1000"> </span><span
class="ecti-1000">normalization</span> <br /><span class="label"><a
id="x1-7013r3"></a></span>b = a.normalized() <span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">returns</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">copy</span><span
class="ecti-1000"> </span><span
class="ecti-1000">of</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000">,</span><span
class="ecti-1000"> </span><span
class="ecti-1000">normalized</span>
</div>
<!--l. 188--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.5 </span> <a
id="x1-80001.2.5"></a>Dot Product</h5>
<!--l. 190--><p class="noindent" >The dot product is, pheraps, the most useful operation that can be applied to 3D
vectors. In the surface, it’s multiple usages are not very obvious, but in depth it can
provide very useful information between two vectors (be it direction or just points in
space).
<!--l. 195--><p class="indent" > The dot product takes two vectors (<span
class="ecti-1000">a </span>and <span
class="ecti-1000">b </span>in the example) and returns a scalar
(single real number):
<div class="center"
>
<!--l. 198--><p class="noindent" >
<!--l. 199--><p class="noindent" ><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">x</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">x</span></sub> <span
class="cmr-10">+ </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">y</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">y</span></sub> <span
class="cmr-10">+ </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">z</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">z</span></sub>
</div>
<!--l. 202--><p class="indent" > The same expressed in code:
<!--l. 205-->
<div class="lstlisting"><span class="label"><a
id="x1-8001r1"></a></span>a = Vector3(...) <br /><span class="label"><a
id="x1-8002r2"></a></span>b = Vector3(...) <br /><span class="label"><a
id="x1-8003r3"></a></span> <br /><span class="label"><a
id="x1-8004r4"></a></span>c = a.x<span
class="cmsy-10">*</span>b.x + a.y<span
class="cmsy-10">*</span>b.y + a.z<span
class="cmsy-10">*</span>b.z <br /><span class="label"><a
id="x1-8005r5"></a></span> <br /><span class="label"><a
id="x1-8006r6"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">using</span><span
class="ecti-1000"> </span><span
class="ecti-1000">built</span><span
class="cmsy-10">-</span><span
class="ecti-1000">in</span><span
class="ecti-1000"> </span><span
class="ecti-1000">dot</span><span
class="ecti-1000">()</span><span
class="ecti-1000"> </span><span
class="ecti-1000">function</span> <br /><span class="label"><a
id="x1-8007r7"></a></span> <br /><span class="label"><a
id="x1-8008r8"></a></span>c = a.dot(b)
</div>
<!--l. 218--><p class="indent" > The dot product presents several useful properties:
<ul class="itemize1">
<li class="itemize">If both <span
class="ecti-1000">a </span>and <span
class="ecti-1000">b </span>parameters to a <span
class="ecti-1000">dot product </span>are direction vectors, dot
product will return positive if both point towards the same direction,
negative if both point towards opposite directions, and zero if they are
orthogonal (one is perpendicular to the other).
</li>
<li class="itemize">If both <span
class="ecti-1000">a </span>and <span
class="ecti-1000">b </span>parameters to a <span
class="ecti-1000">dot product </span>are <span
class="ecti-1000">normalized </span>direction
vectors, then the dot product will return the cosine of the angle between
them (ranging from 1 if they are equal, 0 if they are orthogonal, and -1 if
they are opposed (a == -b)).
</li>
<li class="itemize">If <span
class="ecti-1000">a </span>is a <span
class="ecti-1000">normalized </span>direction vector and <span
class="ecti-1000">b </span>is a point, the dot product will
return the distance from <span
class="ecti-1000">b </span>to the plane passing through the origin, with
normal <span
class="ecti-1000">a (see item about planes)</span>
</li>
<li class="itemize">More uses will be presented later in this tutorial.</li></ul>
<!--l. 236--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.2.6 </span> <a
id="x1-90001.2.6"></a>Cross Product</h5>
<!--l. 238--><p class="noindent" >The <span
class="ecti-1000">cross product </span>also takes two vectors <span
class="ecti-1000">a </span>and <span
class="ecti-1000">b</span>, but returns another vector <span
class="ecti-1000">c </span>that is
orthogonal to the two previous ones.
<div class="center"
>
<!--l. 242--><p class="noindent" >
<!--l. 243--><p class="noindent" ><span
class="cmmi-10">c</span><sub><span
class="cmmi-7">x</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">x</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">z</span></sub> <span
class="cmsy-10">- </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">z</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">y</span></sub>
</div>
<div class="center"
>
<!--l. 246--><p class="noindent" >
<!--l. 247--><p class="noindent" ><span
class="cmmi-10">c</span><sub><span
class="cmmi-7">y</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">z</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">x</span></sub> <span
class="cmsy-10">- </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">x</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">z</span></sub>
</div>
<div class="center"
>
<!--l. 250--><p class="noindent" >
<!--l. 251--><p class="noindent" ><span
class="cmmi-10">c</span><sub><span
class="cmmi-7">z</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">x</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">y</span></sub> <span
class="cmsy-10">- </span><span
class="cmmi-10">a</span><sub><span
class="cmmi-7">y</span></sub><span
class="cmmi-10">b</span><sub><span
class="cmmi-7">x</span></sub>
</div>
<!--l. 254--><p class="indent" > The same in code:
<!--l. 257-->
<div class="lstlisting"><span class="label"><a
id="x1-9001r1"></a></span>a = Vector3(...) <br /><span class="label"><a
id="x1-9002r2"></a></span>b = Vector3(...) <br /><span class="label"><a
id="x1-9003r3"></a></span>c = Vector3(...) <br /><span class="label"><a
id="x1-9004r4"></a></span> <br /><span class="label"><a
id="x1-9005r5"></a></span>c.x = a.x<span
class="cmsy-10">*</span>b.z <span
class="cmsy-10">-</span> a.z<span
class="cmsy-10">*</span>b.y <br /><span class="label"><a
id="x1-9006r6"></a></span>c.y = a.z<span
class="cmsy-10">*</span>b.x <span
class="cmsy-10">-</span> a.x<span
class="cmsy-10">*</span>b.z <br /><span class="label"><a
id="x1-9007r7"></a></span>c.z = a.x<span
class="cmsy-10">*</span>b.y <span
class="cmsy-10">-</span> a.y<span
class="cmsy-10">*</span>b.x <br /><span class="label"><a
id="x1-9008r8"></a></span> <br /><span class="label"><a
id="x1-9009r9"></a></span>// or using the built<span
class="cmsy-10">-</span>in function <br /><span class="label"><a
id="x1-9010r10"></a></span> <br /><span class="label"><a
id="x1-9011r11"></a></span>c = a.cross(b)
</div>
<!--l. 273--><p class="indent" > The <span
class="ecti-1000">cross product </span>also presents several useful properties:
<ul class="itemize1">
<li class="itemize">As mentioned, the resulting vector <span
class="ecti-1000">c </span>is orthogonal to the input vectors <span
class="ecti-1000">a</span>
and <span
class="ecti-1000">b.</span>
</li>
<li class="itemize">Since the <span
class="ecti-1000">cross product </span>is anticommutative, swapping <span
class="ecti-1000">a </span>and <span
class="ecti-1000">b </span>will result
in a negated vector <span
class="ecti-1000">c.</span>
</li>
<li class="itemize">if <span
class="ecti-1000">a </span>and <span
class="ecti-1000">b </span>are taken from two of the segmets <span
class="ecti-1000">AB</span>, <span
class="ecti-1000">BC </span>or <span
class="ecti-1000">CA </span>that form a
3D triangle, the magnitude of the resulting vector divided by 2 is the area
of that triangle.
</li>
<li class="itemize">The direction of the resulting vector <span
class="ecti-1000">c </span>in the previous triangle example
determines wether the points A,B and C are arranged in clocwise or
counter-clockwise order.</li></ul>
<!--l. 287--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">1.3 </span> <a
id="x1-100001.3"></a>Plane</h4>
<!--l. 290--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.3.1 </span> <a
id="x1-110001.3.1"></a>Theory</h5>
<!--l. 292--><p class="noindent" >A plane can be considered as an infinite, flat surface that splits space in two halves,
usually one named positive and one named negative. In regular mathematics, a plane
formula is described as:
<div class="center"
>
<!--l. 296--><p class="noindent" >
<!--l. 297--><p class="noindent" ><span
class="cmmi-10">ax </span><span
class="cmr-10">+ </span><span
class="cmmi-10">by </span><span
class="cmr-10">+ </span><span
class="cmmi-10">cz </span><span
class="cmr-10">+ </span><span
class="cmmi-10">d</span>
</div>
<!--l. 300--><p class="indent" > However, in 3D programming, this form alone is often of little use. For planes to
become useful, they must be in normalized form.
<!--l. 303--><p class="indent" > A normalized plane consists of a <span
class="ecti-1000">normal vector n </span>and a <span
class="ecti-1000">distance d. </span>To normalize
a plane, a vector <span
class="ecti-1000">n </span>and distance <span
class="ecti-1000">d’ </span>are created this way:
<!--l. 307--><p class="indent" > <span
class="cmmi-10">n</span><sub><span
class="cmmi-7">x</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">a</span>
<!--l. 309--><p class="indent" > <span
class="cmmi-10">n</span><sub><span
class="cmmi-7">y</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">b</span>
<!--l. 311--><p class="indent" > <span
class="cmmi-10">n</span><sub><span
class="cmmi-7">z</span></sub> <span
class="cmr-10">= </span><span
class="cmmi-10">c</span>
<!--l. 313--><p class="indent" > <span
class="cmmi-10">d</span><span
class="cmsy-10">′ </span><span
class="cmr-10">= </span><span
class="cmmi-10">d</span>
<!--l. 315--><p class="indent" > Finally, both <span
class="ecti-1000">n </span>and <span
class="ecti-1000">d’ </span>are both divided by the magnitude of n.
<!--l. 318--><p class="indent" > In any case, normalizing planes is not often needed (this was mostly for
explanation purposes), and normalized planes are useful because they can be created
and used easily.
<!--l. 322--><p class="indent" > A normalized plane could be visualized as a plane pointing towards normal <span
class="ecti-1000">n,</span>
offseted by <span
class="ecti-1000">d </span>in the direction of <span
class="ecti-1000">n</span>.
<!--l. 325--><p class="indent" > In other words, take <span
class="ecti-1000">n</span>, multiply it by scalar <span
class="ecti-1000">d </span>and the resulting point will be part
of the plane. This may need some thinking, so an example with a 2D normal vector
(z is 0, so plane is orthogonal to it) is provided:
<!--l. 330--><p class="indent" > Some operations can be done with normalized planes:
<ul class="itemize1">
<li class="itemize">Given any point <span
class="ecti-1000">p</span>, the distance from it to a plane can be computed by
doing: n.dot(p) - d
</li>
<li class="itemize">If the resulting distance in the previous point is negative, the point is
below the plane.
</li>
<li class="itemize">Convex polygonal shapes can be defined by enclosing them in planes (the
physics engine uses this property)</li></ul>
<!--l. 340--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.3.2 </span> <a
id="x1-120001.3.2"></a>Implementation</h5>
<!--l. 342--><p class="noindent" >Godot Engine implements normalized planes by using the <span
class="ecti-1000">Plane </span>class.
<!--l. 346-->
<div class="lstlisting"><span class="label"><a
id="x1-12001r1"></a></span>//creates a plane with normal (0,1,0) and distance 5 <br /><span class="label"><a
id="x1-12002r2"></a></span>p = Plane( Vector3(0,1,0), 5 ) <br /><span class="label"><a
id="x1-12003r3"></a></span>// get the distance to a point <br /><span class="label"><a
id="x1-12004r4"></a></span>d = p.distance( Vector3(4,5,6) )
</div>
<!--l. 355--><p class="noindent" >
<h4 class="subsectionHead"><span class="titlemark">1.4 </span> <a
id="x1-130001.4"></a>Matrices, Quaternions and Coordinate Systems</h4>
<!--l. 357--><p class="noindent" >It is very often needed to store the location/rotation of something. In 2D, it is often
enough to store an <span
class="ecti-1000">x,y </span>location and maybe an angle as the rotation, as that should
be enough to represent any posible position.
<!--l. 362--><p class="indent" > In 3D this becomes a little more difficult, as there is nothing as simple as an angle
to store a 3-axis rotation.
<!--l. 365--><p class="indent" > The first think that may come to mind is to use 3 angles, one for x, one for y and
one for z. However this suffers from the problem that it becomes very cumbersome to
use, as the individual rotations in each axis need to be performed one after another
(they can’t be performed at the same time), leading to a problem called “gimbal
lock”. Also, it becomes impossible to accumulate rotations (add a rotation to an
existing one).
<!--l. 373--><p class="indent" > To solve this, there are two known diferent approaches that aid in solving
rotation, <span
class="ecti-1000">Quaternions </span>and <span
class="ecti-1000">Oriented Coordinate Systems.</span>
<!--l. 378--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.4.1 </span> <a
id="x1-140001.4.1"></a>Oriented Coordinate Systems</h5>
<!--l. 380--><p class="noindent" ><span
class="ecti-1000">Oriented Coordinate Systems </span>(<span
class="ecti-1000">OCS</span>) are a way of representing a coordinate system
inside the cartesian coordinate system. They are mainly composed of 3 Vectors, one
for each axis. The first vector is the <span
class="ecti-1000">x </span>axis, the second the <span
class="ecti-1000">y </span>axis, and the third is the
<span
class="ecti-1000">z </span>axis. The OCS vectors can be rotated around freely as long as they are kept the
same length (as changing the length of an axis changes its cale), and as long as they
remain orthogonal to eachother (as in, the same as the default cartesian system,
with <span
class="ecti-1000">y </span>pointing up, <span
class="ecti-1000">x </span>pointing left and <span
class="ecti-1000">z </span>pointing front, but all rotated
together).
<!--l. 391--><p class="indent" > <span
class="ecti-1000">Oriented Coordinate Systems </span>are represented in 3D programming as a 3x3 matrix,
where each row (or column, depending on the implementation) contains one of the
axis vectors. Transforming a Vector by a rotated OCS Matrix results in the rotation
being applied to the resulting vector. OCS Matrices can also be multiplied to
accumulate their transformations.
<!--l. 397--><p class="indent" > Godot Engine implements OCS Matrices in the <span
class="ecti-1000">Matrix3 </span>class:
<!--l. 400-->
<div class="lstlisting"><span class="label"><a
id="x1-14001r1"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000">create</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">3</span><span
class="ecti-1000">x3</span><span
class="ecti-1000"> </span><span
class="ecti-1000">matrix</span> <br /><span class="label"><a
id="x1-14002r2"></a></span>m = Matrix3() <br /><span class="label"><a
id="x1-14003r3"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000">rotate</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">matrix</span><span
class="ecti-1000"> </span><span
class="ecti-1000">in</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">y</span><span
class="ecti-1000"> </span><span
class="ecti-1000">axis</span><span
class="ecti-1000">,</span><span
class="ecti-1000"> </span><span
class="ecti-1000">by</span><span
class="ecti-1000"> </span><span
class="ecti-1000">45</span><span
class="ecti-1000"> </span><span
class="ecti-1000">degrees</span> <br /><span class="label"><a
id="x1-14004r4"></a></span>m.rotate( Vector3(0,1,0), Math.deg2rad(45) ) <br /><span class="label"><a
id="x1-14005r5"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000">transform</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">vector</span><span
class="ecti-1000"> </span><span
class="ecti-1000">v</span><span
class="ecti-1000"> </span><span
class="ecti-1000">(</span><span
class="ecti-1000">xform</span><span
class="ecti-1000"> </span><span
class="ecti-1000">method</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">used</span><span
class="ecti-1000">)</span> <br /><span class="label"><a
id="x1-14006r6"></a></span>v = Vector3(...) <br /><span class="label"><a
id="x1-14007r7"></a></span>result = m.xform( v )
</div>
<!--l. 412--><p class="indent" > However, in most usage cases, one wants to store a translation together with the
rotation. For this, an <span
class="ecti-1000">origin </span>vector must be added to the OCS, thus transforming it
into a 3x4 (or 4x3, depending on preference) matrix. Godot engine implements this
functionality in the <span
class="ecti-1000">Transform </span>class:
<!--l. 419-->
<div class="lstlisting"><span class="label"><a
id="x1-14010r1"></a></span>t = Transform() <br /><span class="label"><a
id="x1-14011r2"></a></span>//rotate the transform in the y axis, by 45 degrees <br /><span class="label"><a
id="x1-14012r3"></a></span>t.rotate( Vector3(0,1,0), Math.deg2rad(45) ) <br /><span class="label"><a
id="x1-14013r4"></a></span>//translate the transform by 5 in the z axis <br /><span class="label"><a
id="x1-14014r5"></a></span>t.translate( Vector3( 0,0,5 ) ) <br /><span class="label"><a
id="x1-14015r6"></a></span>//transform a vector v (xform method is used) <br /><span class="label"><a
id="x1-14016r7"></a></span>v = Vector3(...) <br /><span class="label"><a
id="x1-14017r8"></a></span>result = t.xform( v )
</div>
<!--l. 431--><p class="indent" > Transform contains internally a Matrix3 “basis” and a Vector3 “origin” (which can
be modified individually).
<!--l. 435--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.4.2 </span> <a
id="x1-150001.4.2"></a>Transform Internals</h5>
<!--l. 437--><p class="noindent" >Internally, the xform() process is quite simple, to apply a 3x3 transform to a vector,
the transposed axis vectors are used (as using the regular axis vectors will result on
an inverse of the desired transform):
<!--l. 442-->
<div class="lstlisting"><span class="label"><a
id="x1-15001r1"></a></span>m = Matrix3(...) <br /><span class="label"><a
id="x1-15002r2"></a></span>v = Vector3(..) <br /><span class="label"><a
id="x1-15003r3"></a></span>result = Vector3(...) <br /><span class="label"><a
id="x1-15004r4"></a></span> <br /><span class="label"><a
id="x1-15005r5"></a></span>x_axis = m.get_axis(0) <br /><span class="label"><a
id="x1-15006r6"></a></span>y_axis = m.get_axis(1) <br /><span class="label"><a
id="x1-15007r7"></a></span>z_axis = m.get_axis(2) <br /><span class="label"><a
id="x1-15008r8"></a></span> <br /><span class="label"><a
id="x1-15009r9"></a></span>result.x = Vector3(x_axis.x, y_axis.x, z_axis.x).dot(v) <br /><span class="label"><a
id="x1-15010r10"></a></span>result.y = Vector3(x_axis.y, y_axis.y, z_axis.y).dot(v) <br /><span class="label"><a
id="x1-15011r11"></a></span>result.z = Vector3(x_axis.z, y_axis.z, z_axis.z).dot(v) <br /><span class="label"><a
id="x1-15012r12"></a></span> <br /><span class="label"><a
id="x1-15013r13"></a></span>// is the same as doing <br /><span class="label"><a
id="x1-15014r14"></a></span> <br /><span class="label"><a
id="x1-15015r15"></a></span>result = m.xform(v) <br /><span class="label"><a
id="x1-15016r16"></a></span> <br /><span class="label"><a
id="x1-15017r17"></a></span>// if m this was a Transform(), the origin would be added <br /><span class="label"><a
id="x1-15018r18"></a></span>// like this: <br /><span class="label"><a
id="x1-15019r19"></a></span> <br /><span class="label"><a
id="x1-15020r20"></a></span>result = result + t.get_origin()
</div>
<!--l. 468--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.4.3 </span> <a
id="x1-160001.4.3"></a>Using The Transform</h5>
<!--l. 470--><p class="noindent" >So, it is often desired apply sucessive operations to a transformation. For example,
let’s a assume that there is a turtle sitting at the origin (the turtle is a logo reference,
for those familiar with it). The <span
class="ecti-1000">y </span>axis is up, and the the turtle’s nose is pointing
towards the <span
class="ecti-1000">z </span>axis.
<!--l. 476--><p class="indent" > The turtle (like many other animals, or vehicles!) can only walk towards the
direction it’s looking at. So, moving the turtle around a little should be something
like this:
<!--l. 481-->
<div class="lstlisting"><span class="label"><a
id="x1-16001r1"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">turtle</span><span
class="ecti-1000"> </span><span
class="ecti-1000">at</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">origin</span> <br /><span class="label"><a
id="x1-16002r2"></a></span>turtle = Transform() <br /><span class="label"><a
id="x1-16003r3"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">turtle</span><span
class="ecti-1000"> </span><span
class="ecti-1000">will</span><span
class="ecti-1000"> </span><span
class="ecti-1000">walk</span><span
class="ecti-1000"> </span><span
class="ecti-1000">5</span><span
class="ecti-1000"> </span><span
class="ecti-1000">units</span><span
class="ecti-1000"> </span><span
class="ecti-1000">in</span><span
class="ecti-1000"> </span><span
class="ecti-1000">z</span><span
class="ecti-1000"> </span><span
class="ecti-1000">axis</span> <br /><span class="label"><a
id="x1-16004r4"></a></span>turtle.translate( Vector3(0,0,5) ) <br /><span class="label"><a
id="x1-16005r5"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">turtle</span><span
class="ecti-1000"> </span><span
class="ecti-1000">eyes</span><span
class="ecti-1000"> </span><span
class="ecti-1000">a</span><span
class="ecti-1000"> </span><span
class="ecti-1000">lettuce</span><span
class="ecti-1000"> </span><span
class="ecti-1000">3</span><span
class="ecti-1000"> </span><span
class="ecti-1000">units</span><span
class="ecti-1000"> </span><span
class="ecti-1000">away</span><span
class="ecti-1000">,</span><span
class="ecti-1000"> </span><span
class="ecti-1000">will</span><span
class="ecti-1000"> </span><span
class="ecti-1000">rotate</span><span
class="ecti-1000"> </span><span
class="ecti-1000">45</span><span
class="ecti-1000"> </span><span
class="ecti-1000">degrees</span><span
class="ecti-1000"> </span><span
class="ecti-1000">right</span> <br /><span class="label"><a
id="x1-16006r6"></a></span>turtle.rotate( Vector3(0,1,0), Math.deg2rad(45) ) <br /><span class="label"><a
id="x1-16007r7"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">turtle</span><span
class="ecti-1000"> </span><span
class="ecti-1000">approaches</span><span
class="ecti-1000"> </span><span
class="ecti-1000">the</span><span
class="ecti-1000"> </span><span
class="ecti-1000">lettuce</span> <br /><span class="label"><a
id="x1-16008r8"></a></span>turtle.translate( Vector3(0,0,5) ) <br /><span class="label"><a
id="x1-16009r9"></a></span><span
class="ecti-1000">//</span><span
class="ecti-1000"> </span><span
class="ecti-1000">happy</span><span
class="ecti-1000"> </span><span
class="ecti-1000">turtle</span><span
class="ecti-1000"> </span><span
class="ecti-1000">over</span><span
class="ecti-1000"> </span><span
class="ecti-1000">lettuce</span><span
class="ecti-1000"> </span><span
class="ecti-1000">is</span><span
class="ecti-1000"> </span><span
class="ecti-1000">at</span> <br /><span class="label"><a
id="x1-16010r10"></a></span>print(turtle.get_origin())
</div>
<!--l. 496--><p class="indent" > As can be seen, every new action the turtle takes is based on the previous one it
took. Had the order of actions been different and the turtle would have never reached
the lettuce.
<!--l. 500--><p class="indent" > Transforms are just that, a mean of “accumulating” rotation, translation, scale,
etc.
<!--l. 504--><p class="noindent" >
<h5 class="subsubsectionHead"><span class="titlemark">1.4.4 </span> <a
id="x1-170001.4.4"></a>A Warning about Numerical Precision</h5>
<!--l. 506--><p class="noindent" >Performing several actions over a transform will slowly and gradually lead to
precision loss (objects that draw according to a transform may get jittery, bigger,
smaller, skewed, etc). This happens due to the nature of floating point numbers. if
transforms/matrices are created from other kind of values (like a position and
some angular rotation) this is not needed, but if has been accumulating
transformations and was never recreated, it can be normalized by calling the
.orthonormalize() built-in function. This function has little cost and calling it every
now and then will avoid the effects from precision loss to become visible.
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