<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html > <head><title></title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)"> <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)"> <!-- html --> <meta name="src" content="tutorial.tex"> <meta name="date" content="2009-10-07 00:28:00"> <link rel="stylesheet" type="text/css" href="tutorial.css"> </head><body > <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. </body></html>