\n<\/i> Don’t hesitate to bring more references in the comments.<\/div>\n
Source code<\/h2>\n
I use glm library to manipulate matrix and quaternion. It’s free, it’s only headers (no lib or dll) and it’s cross-platform.<\/p>\n
Download glm<\/a><\/p>\n Finally, I use macros EXPECT_EQ and others from Google Test<\/a>.<\/p>\n All following examples are inside my GitHub https:\/\/github.com\/Rominitch\/myBlogSource\/<\/a>, see Matrix_Quaternion.<\/p>\n In the following examples, I will use the following writing convention:<\/p>\n <\/span> <\/span> So we can write: I don’t know if it is a “true” mathematical writing convention but it’s similar to code and c++\/glm order.<\/p>\n For simplification reasons and to have more readable diagrams, we work in Z=0 plane. I have an orthonormal system O with its X axis<\/span> and its Y axis<\/span>, Z is oriented toward us. The transformation matrices containing homogeneous coordinates (that we call matrices afterward) allow us to apply a geometric transformation (more or less complexe) to point or vector.<\/p>\n I suppose you either know how to manipulate (basically realize sum\/multiplication\/transpose\/invert) or your software\/library can do it for you !<\/p>\n Here, you can find the major forms of transformation matrix we want to use: We will start with a rather naive approach. To start with we will try to translate a 3D point. We can easily read Finally, we retrieve our point B but …<\/p>\n Now things are getting more difficult because we don’t actually apply a translation. It’s more like changing our point of view. We are looking at the same object but from a different place.<\/p>\n Actually, matrices allow to change the coordinate system.<\/p>\n I call The real form is: <\/span> <\/span> This is the “actual” diagram of what happened: In fact, our translation matrix will create another basis M with x,y,z vectors similar to O basis. We notice that:<\/p>\n There is sill one last question we need to answer: If B coordinates are written in O basis, in which basis is A written ?<\/p>\n The answer is simple: inside M basis.<\/p>\n Finally, we defined both O and M system. A point is defined in M system and B in O. Our matrix realized a transformation from M to O system. We have just added an important notion: the application way. To conclude:<\/p>\n For what follows, we will create all our matrix in this way: current to up because they are more intuitive to create.<\/p>\n The rotation matrix allows us to define the basis of our system. Each vector is normalized before being used inside the matrix. It’s very simple to write rotation matrix of Reading coordinate, we have I integrate normalized vectors into the matrix and I have:<\/p>\n Let’s try to play with this example:\u00a0 <\/span> <\/span> Perfect ! We did it !<\/p>\n Some remarks:<\/p>\n On complex model (many points), rotation will give you:<\/p>\n Here, we want to “stretch” our object into each dimension. For that, we applied on each basis’ vector a scaling factor. In this example, we try to reduce or system by two in all dimensions. To conclude, let’s try it on our robot.<\/p>\n Some remarks:<\/p>\n To conclude this first part, we saw than matrix creation is not really complicated despite the fact that rotation demands some computation but analize is accessible (during debug phase).<\/p>\n Generally, the most important is to quickly\u00a0 apply all these operations to our billion points. So the main idea is to create only one matrix with allthe operations inside. First we try to compose each matrix which define a complete system (translation \/ rotation \/ homothetic). So to simplify this post, I will call them: matrices HRT. Ok, first, we compute composition.<\/p>\n If we use it on <\/span> <\/span> Now, we can “translate” complex system to matrix, but we need to compose many systems based on hierarchy. So we take another example.<\/p>\n Previously, I told you rotation matrix turn around the origin of a local system. Sometime we need to change it: I will show you the methodology.<\/p>\n Finally, we realize: one translation + one rotation + one translation. Let’s try: Using this example: Start with composition matrix:<\/p>\n Now, try it on points<\/p>\n <\/span> <\/span> I leave you the care of computing C and you see we have a good result.<\/p>\n Our solution is easy to understand and intuitive, moreover it demands only one multiplication.<\/p>\n For your information, one last diagram: To conclude, Our HRT matrices can be composed and cascaded: this is an important subject which helps us when we try to use them into scene graph.<\/p>\n We saw :<\/p>\n Here we have all information to use them into scene graph. Thanks for reading!<\/p>\n Don’t hesitate to shared, post a comment\/fix in order to make it a dynamic tools !<\/p>\n Wikipedia : Produit matriciel<\/a> Hi everyone, today I want to speak about math, simply if possible, because the topic is about matrix & quaternion. The main goal of this series of posts is to use them correctly inside an applied case: the scene graph and how to compute modelview-projection. Preface I want my approach to be more recreational than<\/p>\nNotation<\/h2>\n
\n
Simple notation<\/h4>\n
![\"{\color{myB}r} = {\color{myR}m} \times {\color{myG}v}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-ebdb6d1b123151ff52ff7075b1b72686_l3.png\" "\\"Rendered")
<\/div>\nExtend notation<\/h4>\n
![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-2fa05004f84263fecda803686527fd67_l3.png\" "\\"Rendered")
<\/p><\/div>\nEquivalent source code<\/h4>\n
glm::mat4 matrix(0.0f, 1.0f, 0.0f, 0.0f,\r\n 1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\nglm::vec4 vector(2.0f, 3.0f, 4.0f, 1.0f);\r\nglm::vec4 result = matrix * vector;<\/pre><\/div>
![\"{\color{myB}mvp} = {\color{myR}p} \times ({\color{myG}v} \times {\color{myY}m})\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-6ed561f7dfca4198486993859de11525_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-763bfde6e0efa9082b5c9e7057be61e8_l3.png\" "\\"Rendered")
<\/div>\nglm::mat4 m(0.5f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.5f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 0.5f, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\nglm::mat4 v(0.0f, 1.0f, 0.0f, 0.0f,\r\n 1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\nglm::mat4 p(1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 1.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 1.0f, 2.0f, 3.0f, 1.0f);\r\n\r\nglm::mat4 result = p * v * m;\r\nglm::mat4 mvp(0.0f, 0.5f, 0.0f, 0.0f,\r\n 0.5f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 0.5f, 0.0f,\r\n 1.0f, 2.0f, 3.0f, 1.0f);\r\nEXPECT_EQ(mvp, result);\r\nEXPECT_NE(mvp, m * v * p);\r\n<\/pre><\/div>
\nWe stay in 3D in our logic as well as in our formula.<\/p>\n
\nUnits depend on the size of x and y vectors.
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/classic_repere-290x216.png\")
\nIn short, it’s classic.<\/p>\nTransformation matrices<\/h1>\n
Remind: major forms<\/h2>\n
\nTranslation T<\/h4><\/div><\/p>\n
Rotation R<\/h4><\/div>\n
Homothetic or Scaling S<\/h4><\/div>
![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-07933042648f32d4bdba63201941633f_l3.png\" "\\"Rendered")
<\/div><\/div>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5456a53614b4a3db3d845949ea88f94f_l3.png\" "\\"Rendered")
<\/div><\/div>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-e5c5294815939e749f4b7fd3027d6f5e_l3.png\" "\\"Rendered")
<\/div><\/div>Problem 1: What are transformation matrix ? and how to make them ?<\/h2>\n
Translation and matrix representation<\/h3>\n
\nOn this diagram, I want to translate my point A to B.
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/translation-1-300x228.png\"){kind=spacer}
<\/p>\n![\"{\color{myB}A(1, 1, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-3bdc660f383432c74d2321422ddf7cc0_l3.png\" "\\"Rendered")
and ![\"{\color{myO}B(3, 2, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-40460bb976a15091ee90b612c868baed_l3.png\" "\\"Rendered")
so translation vector is ![\"{\color{myR}\vec{AB}(2, 1, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-548587ebbc13ec40e3161ed60dc5d60c_l3.png\" "\\"Rendered")
.
\nWe substitute the “t” values of the matrix by this vector and compute the multiplication with A.<\/p>\n![\"\begin{array}{c c}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-177a851540e1939acd20b7e9d03ea7c8_l3.png\" "\\"Rendered")
<\/div>glm::vec4 b( 3.0f, 2.0f, 0.0f, 1.0f);\r\n\r\nglm::vec4 a( 1.0f, 1.0f, 0.0f, 1.0f);\r\nglm::mat4 t( 1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 1.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 2.0f, 1.0f, 0.0f, 1.0f );\r\nEXPECT_EQ( b, t * a );\r\n\r\nglm::mat4 t2( 1.0f ); \/\/ Identity\r\nt2 = glm::translate( t2, glm::vec3( 2.0f, 1.0f, 0.0f ) );\r\nEXPECT_EQ( b, t2 * a );\r\n<\/pre><\/div>
\n
![\"\mathscr{M}_{X\vec{x}\vec{y}\vec{z}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-9acfdb439811e942bc31780c78f8ac04_l3.png\" "\\"Rendered")
system, a system defined by X origin and ![\"\vec{x}\vec{y}\vec{z}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-6d07f41815847b6233f1b347bb8a18e5_l3.png\" "\\"Rendered")
basis.<\/p>\n
\n![\"\begin{align*}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-2e3a0321a8a6a2bb1a17110cf01101bb_l3.png\" "\\"Rendered")
<\/p>\n<\/div><\/div><\/p>\n
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/repere-1-300x229.png\")
<\/p>\n\n
![\"\vec{OA} = \vec{MB}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-fef4c241ffabd74a2c445cf04c92d465_l3.png\" "\\"Rendered")
so the initial drawing (here just a point) doesn’t move in relation to basis.<\/li>\n![\"O\vec{x}\vec{y}\vec{z}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-d8902e1585c503437d146cbd8bdebb79_l3.png\" "\\"Rendered")
basis.<\/li>\n<\/ul>\n
\nIt has its own system![\"\mathscr{S}_{R\vec{s}\vec{t}\vec{r}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-dd8a1041431498a8e3e4cb5f794533bd_l3.png\" "\\"Rendered")
and a point ![\"{\color{myG}E(1, 1, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-02248713abd724916eb29955a38b0da3_l3.png\" "\\"Rendered")
representing the eye.
<\/div>\n![\"\mathscr{S}_{O\vec{x}\vec{y}\vec{z}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-0acdae408810d8e971b4a3dfb2881588_l3.png\" "\\"Rendered")
.
\nand another ![\"\mathscr{S}_{M\vec{u}\vec{v}\vec{w}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-78f6c055d6297564b1a76682f25e7932_l3.png\" "\\"Rendered")
which hierarchically depends from O because point M is defined in O system.
<\/div>\n), I will transform my point E to A for O system or B for M system.
<\/div>![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/robo1-1.png\")
<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/robo2-290x220.png\")
<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/robo3-290x220.png\")
<\/div>
\n is upper system of because M coordinates are defined in : we have a hierarchy notion.<\/p>\n
\nIn order to change it, we need to invert the matrix. Indeed, all the previous transformation matrices are invertible.<\/p>\n\n
\nWorld is mainly the top of the hierarchy.<\/div>\nThe rotation<\/h3>\n
\nWe have the following matrix:<\/p>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-080f6d2fbf2aef326de392b1ac7fc947_l3.png\" "\\"Rendered")
<\/div><\/div>\n![\"\frac{PI}{4}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-c696d1c7f980d95b8ec929b5ae774367_l3.png\" "\\"Rendered")
around ![\"\vec{z}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-fe25dcb2a0e83df97cac680b41305d43_l3.png\" "\\"Rendered")
.
\nWe start with this diagram:<\/p>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotation_without_norm-1-300x192.png\")
<\/p>\n![\"\color{myM}{\vec{x} = (1, 1, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-f2e361d60144419a94612f77b8cca1e1_l3.png\" "\\"Rendered")
, ![\"\color{myY}{\vec{y} = (-1, 1, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-97efbf16ed14e427fb3147368deb64d3_l3.png\" "\\"Rendered")
\u00a0and ![\"\color{myB}{\vec{z} = (0, 0, 1)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-884c8afd01570f08dbb8c26306ae71ba_l3.png\" "\\"Rendered")
.
\nI define ![\" C = \frac{1}{\sqrt{2}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-69766f98174d4d0603c07cff0544cf51_l3.png\" "\\"Rendered")
soit ![\"\frac{1}{vector length}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-75b0b2a577972fb3271187e00abada15_l3.png\" "\\"Rendered")
which allows me to normalize vectors.<\/p>\n![\" C = cos(\frac{PI}{4}) = sin(\frac{PI}{4}) = \frac{1}{\sqrt{2}}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-4daf823960f4c3a90149eccfe804856a_l3.png\" "\\"Rendered")
. For those who know their rotation matrix around ![\"\vec{Z}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-76af122d9ba98d937403aa24a29883b7_l3.png\" "\\"Rendered")
, by hearth sin() et cos() are here.<\/div>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-424aa16748423b6357f74743405002c1_l3.png\" "\\"Rendered")
<\/div>
\nconst float pi_4 = 0.785398163397448309616f;\r\nconst float C = 1.0f \/ sqrt(2.0f);\r\n\/\/ Made hand matrix\r\nglm::mat4 r( C, C, 0.0f, 0.0f,\r\n -C, C, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\n \r\n\/\/ Or using glm\r\nglm::mat4 r1(1.0f); \/\/ Identity\r\nr1 = glm::rotate(r1, pi_4, glm::vec3(0.0f, 0.0f, 1.0f));\r\n \r\nEXPECT_EQ(r, r1);\r\n<\/pre><\/div>
![\"\color{myR}{A = (\frac{2}{3}, \frac{1}{3}, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-acb32d2f47d7493b7fd7205c2a0c28fc_l3.png\" "\\"Rendered")
and ![\"\color{myB}{B = (\frac{C}{3}, C, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5d3e44d8a8406551ca06e68e6b859e5e_l3.png\" "\\"Rendered")
(it is simple enough to compute using Pythagore theorem only and in noticing than OA = y coordinate of point B)<\/p>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotation_point-1-300x215.png\")
<\/p>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-4b45b3956ba66dced3792ae94e7e9690_l3.png\" "\\"Rendered")
<\/p><\/div>
\nconst float C = 1.0f \/ sqrt(2.0f);\r\nglm::mat4 r( C, C, 0.0f, 0.0f,\r\n -C, C, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\n\r\nglm::vec4 a(2.0f\/3.0f, 1.0f\/3.0f, 0.0f, 1.0f);\r\nglm::vec4 b( C\/3.0f, C, 0.0f, 1.0f);\r\n\r\nEXPECT_EQ(b, r * a);\r\n<\/pre><\/div>
\n
system (as expected).<\/li>\n<\/ul>\n<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotation-1-300x219.png\")
\nAnother question is raised: how to turn around another point ? Ok … I leave it for now but you have the reply at the end of the post.<\/p>\nand finally homothetic transformation<\/h4>\n
\nEach scaling factor is defined in ![\"\mathbb{R}_{+}^{*}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-2e13eb3923d259cfa740699955ee93a3_l3.png\" "\\"Rendered")
.
\nWhen superior to 1 : we increase, and between 0 and 1 : we reduce.<\/p>\n
\nThis time we take two points in order to see what happens.<\/p>\n\n
![\"\"](\"https:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/scaling.png\")
<\/p>\n![\"\begin{array}{c c}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-66ef460c919c04e785cb752f83e80391_l3.png\" "\\"Rendered")
<\/div>
\nglm::vec3 h(0.5f, 0.5f, 0.5f);\r\nglm::mat4 m(h.x, 0.0f, 0.0f, 0.0f,\r\n 0.0f, h.y, 0.0f, 0.0f,\r\n 0.0f, 0.0f, h.z, 0.0f,\r\n 0.0f, 0.0f, 0.0f, 1.0f);\r\n\r\nglm::mat4 m1(1.0f); \/\/ Identity\r\nm1 = glm::scale(m1, h);\r\nEXPECT_EQ(m1, m);\r\n\r\nglm::vec4 a(2.0f\/3.0f, 1.0f\/3.0f, 0.0f, 1.0f);\r\nglm::vec4 b(1.0f, 1.0f\/3.0f, 0.0f, 1.0f);\r\n\r\nglm::vec4 c(1.0f\/3.0f, 1.0f\/6.0f, 0.0f, 1.0f);\r\nglm::vec4 d(1.0f\/2.0f, 1.0f\/6.0f, 0.0f, 1.0f);\r\n\r\nEXPECT_EQ(c, m * a);\r\nEXPECT_EQ(d, m * b);\r\n<\/pre><\/div>
![\"\"](\"https:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/scaling_object.png\")
<\/p>\n\n
system.<\/li>\n![\"\frac{2}{3}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-d3b766245d69c7b7ccf6b63420b0fa46_l3.png\" "\\"Rendered")
of X vector and D at the end of the vector.<\/li>\n<\/ul>\n<\/div>\nProblem 2: Composition<\/h2>\n
\nTo create them we use composition which is a multiplication between matrices.<\/p>\n
\nMATRIX PRODUCT IS NOT COMMUTATIVE: ![\" m \times n \neq n \times m\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5aba3e49bd192e3b8db5d9fdfa5a8d01_l3.png\" "\\"Rendered")
!<\/div>\n
\nSo, the order of the operations are very important but the rules are simple:<\/p>\n![\" \color{myM}{Composition} = \color{myB}{Translation} \times (\color{myR}{Rotation} \times \color{myG}{Homothetic}) \"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-2ca4935a68f2e981094128cd03bfce4f_l3.png\" "\\"Rendered")
<\/p>\n
\nAs usual, I will take a example to see what happens.
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/composition_simple-300x194.png\")
<\/p>\n![\"\begin{array}{c c c}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-b2b819e885bdc26cfd9608a72cbff4ed_l3.png\" "\\"Rendered")
<\/div>
\nconst float PI = 3.14159265358979323846f;\r\n\r\nglm::mat4 s = glm::scale(glm::mat4(1.0f), glm::vec3(0.5f, 0.5f, 0.5f));\r\nglm::mat4 r = glm::rotate(glm::mat4(1.0f), PI*0.5f, glm::vec3(0.0f, 0.0f, 1.0f));\r\nglm::mat4 t = glm::translate(glm::mat4(1.0f), glm::vec3(11.0f\/6.0f, 0.5f, 0.0f));\r\n \r\nglm::mat4 m(0.0f, 0.5f, 0.0f, 0.0f,\r\n -0.5f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 0.5f, 0.0f,\r\n 11.0f\/6.0f, 0.5f, 0.0f, 1.0f);\r\nglm::mat4 c = t * r * s;\r\n \r\n\/\/ Near expected\r\nfor(int rawID=0; rawID < 4; ++rawID)\r\n{\r\n for(int columnID=0; columnID < 4; ++columnID)\r\n {\r\n EXPECT_NEAR(m[rawID][columnID], c[rawID][columnID], 1e-6f);\r\n }\r\n}\r\n<\/pre><\/div>![\"\color{myR}{A}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-dbb19a78e088df59e0ee9268cff9e26b_l3.png\" "\\"Rendered")
.<\/p>\n![\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5bc271507db4401c3132b0182e7069db_l3.png\" "\\"Rendered")
<\/p><\/div>
\nglm::mat4 m(0.0f, 0.5f, 0.0f, 0.0f,\r\n -0.5f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 0.5f, 0.0f,\r\n 11.0f\/6.0f, 0.5f, 0.0f, 1.0f);\r\nglm::vec4 a(2.0f\/3.0f, 1.0f\/3.0f, 0.0f, 1.0f);\r\nglm::vec4 b(5.0f\/3.0f, 5.0f\/6.0f, 0.0f, 1.0f);\r\n\r\nglm::vec4 mul = m * a;\r\n \r\n\/\/ Near expected\r\nfor(int columnID=0; columnID < 4; ++columnID)\r\n{\r\n EXPECT_NEAR(b[columnID], mul[columnID], 1e-6f);\r\n}\r\n<\/pre><\/div>Turn around another point<\/h2>\n
![\"\vec{BO}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-84e29a1e75d28d8525aa21a3d0fd0209_l3.png\" "\\"Rendered")
.<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotate1.png\")
<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotate2.png\")
<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotate3-206x220.png\")
<\/div>![\"\vec{OB}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-518be52fc2717b229c6fe4b45cbf209d_l3.png\" "\\"Rendered")
.<\/div>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotate4-290x220.png\")
<\/div>\n
\nWe see how to make HRT matrix but can we combine more of them?<\/p>\n
\n![\" \color{myM}{Composition} = \color{myB}{Translation\vec{OP}} \times (\color{myR}{Rotation} \times \color{myG}{Translation\vec{PO}}) \"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-9a3906b5e1aa679dc10eb8e04be02e6c_l3.png\" "\\"Rendered")
<\/p>\n![\"{\color{myG}A(1, 2, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-e9130d3769137e6ff990df179bc1e1f7_l3.png\" "\\"Rendered")
and ![\"{\color{myB}B(\frac{4}{3}, \frac{4}{3}, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-e223b036f6a0c22b5f08a6b79ca49896_l3.png\" "\\"Rendered")
some points of our object, we try to apply a rotation of ![\"\frac{\pi}{2}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-b40d4729101107a208fc1dfcffeb5002_l3.png\" "\\"Rendered")
around A.
\nWe expect ![\"{\color{myG}C(1, 2, 0) = A}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-cceeb0aebf6e1ebb028eaa72a383406f_l3.png\" "\\"Rendered")
and ![\"{\color{myO}D(\frac{5}{3}, \frac{7}{3}, 0)}\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-84f0caa4f354dd1248487a49d35419d7_l3.png\" "\\"Rendered")
<\/p>\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotatePoint1.png\")
<\/div>
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/rotatePoint2.png\")
<\/div>![\"\begin{array}{c c c}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-1ca3e559c80ea733a8bdfaee263f9115_l3.png\" "\\"Rendered")
<\/div>
\nconst float PI = 3.14159265358979323846f;\r\n\r\nglm::mat4 m( 0.0f, 1.0f, 0.0f, 0.0f,\r\n -1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 3.0f, 1.0f, 0.0f, 1.0f );\r\n\r\nglm::mat4 op = glm::translate( glm::mat4( 1.0f ), glm::vec3( 1.0f, 2.0f, 0.0f ) );\r\nglm::mat4 po = glm::inverse( op );\r\nglm::mat4 r = glm::rotate( glm::mat4( 1.0f ), PI*0.5f, glm::vec3( 0.0f, 0.0f, 1.0f ) );\r\n\r\nglm::mat4 c = op * r * po;\r\n\r\n\/\/ Near expected\r\nfor( int rawID=0; rawID < 4; ++rawID )\r\n{\r\n for( int columnID=0; columnID < 4; ++columnID )\r\n {\r\n EXPECT_NEAR( m[rawID][columnID], c[rawID][columnID], 1e-6f );\r\n }\r\n}\r\n<\/pre><\/div>![\"\begin{align*}](\"http:\/\/mouca.fr\/wordpress\/wp-content\/ql-cache\/quicklatex.com-7a3bc6d58bb560df2973dbb9f2e25b1a_l3.png\" "\\"Rendered")
<\/p><\/div>
\nglm::mat4 m( 0.0f, 1.0f, 0.0f, 0.0f,\r\n -1.0f, 0.0f, 0.0f, 0.0f,\r\n 0.0f, 0.0f, 1.0f, 0.0f,\r\n 3.0f, 1.0f, 0.0f, 1.0f );\r\n\r\nglm::vec4 a( 1.0f, 2.0f, 0.0f, 1.0f );\r\nglm::vec4 b( 4.0f\/3.0f, 4.0f\/3.0f, 0.0f, 1.0f );\r\nglm::vec4 c( 1.0f, 2.0f, 0.0f, 1.0f );\r\nglm::vec4 d( 5.0f\/3.0f, 7.0f\/3.0f, 0.0f, 1.0f );\r\n\r\nEXPECT_EQ( c, m * a );\r\n\r\nglm::vec4 computed = m * b;\r\n\/\/ Near expected\r\nfor( int columnID=0; columnID < 4; ++columnID )\r\n{\r\n EXPECT_NEAR( d[columnID], computed[columnID], 1e-6f );\r\n}\r\n<\/pre><\/div>
\n![\"\"](\"http:\/\/mouca.fr\/wordpress\/wp-content\/uploads\/2018\/03\/all_rotate.png\")
<\/p>\nConclusion<\/h2>\n
\n
\nBut, I want to speak about quaternions before: ! Wait for the translation !<\/a>
\nIf you don’t need them: ! Soon !<\/a><\/p>\nReferences<\/h2>\n
\nWikipedia : Transformation matrix<\/a>
\nWikipedia : Matrice de passage<\/a>
\nwww.opengl-tutorial.org – Tutorial 3 : Matrices<\/a>
\nLes transformations g\u00e9om\u00e9triques du plan<\/a>
\nUniversity of Edinburgh – Transformations – Taku Komura<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"