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• A system in a space $latex \mathb{X}$ is a collection of points that may move over time.

A system in a $latex \mathbb{R}^n$ is said to be a rigid body if two things are true:

• distance between any two points in the body never changes over time
• the “orientation” of the body is preserved over time

A mechanism is a finite collection of rigid bodies $latex (b_1, \ldots, b_k)$ such that between $latex b_i$ and $latex b_{i+1}$ (for $latex 1 \leq i < k$) there is a joint of some kind connecting the two bodies.

• The type of the joint determines the relative motion of the bodies connected by the joint. For example, when two bodies are connected via a prismatic joint, only translation along some axis is permitted between the two bodies. In the case of a revolute joint, only rotation about some axis is allowed.

• The important thing to note is that the only kind of relative motion that joints permit is rigid motion. A rigid transformation of $latex \mathbb{R}^n$
is a map $latex x \mapsto Ax + d$ for some orthogonal map $latex A$ with $latex det A > 0$ and some $latex d \in \mathbb{R}^n$.

It’s seems helpful to note one thing: the class of all isometries on $latex \mathbb{R}^n$ (i.e. all the maps $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex |f(x) - f(y)| = |x - y|$ for all $latex x, y \in \mathbb{R}^n$) is a strict superset of the class of rigid transformations.

• It can be proved that every isometry on $latex \mathbb{R}^n$ is necessarily affine. So we proceed by studying the collection of affine isometries.

• The collection of affine isometries is exactly the set of all compositions $latex t \circ k$ where $latex t$ is a translation and $latex k$ is a central isometry. Proof: (…)

• But one way to characterize the collection of all central isometries on $latex \mathbb{R}^n$ is as the collection of general orthogonal maps on $latex \mathbb{R}^n$.

• It can be shown that every general orthogonal map is necessarily linear.

• Hence the central isometries of $latex \mathbb{R}^n$ are exactly all the elements of $latex O(n)$, the group of linear orthogonal maps, also known as the orthogonal group.

• So not every isometry is a rigid transformation (but every rigid transformation is an isometry).

Note that an isometry is an orthogonal map followed by a translation, however a rigid transformation is a rotation followed by a translation. So the rigid motions are what you get when you start with the isometries and throw out reflections (and glide reflections). throwing out reflections, since they do not preserve orientation.

• An affine isometry is an affine transformation which is also an isometry.

A translation is any displacement of $latex \mathbb{R}^n$ by a single fixed vector, for example $latex x \mapsto x + d$.

A central isometry is any isometry of $latex \mathbb{R}^n$ that fixes zero. That is, an $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex f(0) = 0$.

• The collection of affine isometries on $latex \mathbb{R}^n$ is exactly the collection of compositions $latex t \circ k$ where $latex t$ is a translation of $latex \mathbb{R}^n$ and $latex k$ is a central isometry.

Proof: blah blah blah math goes here. $latex \Box$

• A general orthogonal map is a not necessarily linear function $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex f(x) \cdot f(y) = x \cdot y$ for all $latex x, y \in \mathbb{R}^n$.

• If $latex V$ is any vector space, the group $latex GL(V)$ of all bijective linear maps $latex V \to V$ is called the general linear group on $latex V$. In the case of $latex V = \mathbb{R}^n$, we write $latex GL(n)$ instead.

$latex O(n)$ can be defined to be

$latex O(n) := \{ f \in GL(n) : \forall x, y \in \mathbb{R}^n, x \cdot y = f(x) \cdot f(y)$.

• A glide reflection is the composition of a translation and a reflection.

{"cards":[{"_id":"4dcc1ccf289482c4ce000055","treeId":"4dcbf502289482c4ce00004f","seq":1374071,"position":0.5,"parentId":null,"content":"A **system** in a space $latex \\mathb{X}$ is a collection of points that may move over time. \n\nA system in a $latex \\mathbb{R}^n$ is said to be a **rigid body** if two things are true:\n\n - distance between any two points in the body never changes over time\n - the \"orientation\" of the body is preserved over time\n\nA **mechanism** is a finite collection of rigid bodies $latex (b_1, \\ldots, b_k)$ such that between $latex b_i$ and $latex b_{i+1}$ (for $latex 1 \\leq i < k$) there is a joint of some kind connecting the two bodies."},{"_id":"4dcc2f63289482c4ce000056","treeId":"4dcbf502289482c4ce00004f","seq":1346606,"position":0.75,"parentId":null,"content":"The type of the joint determines the relative motion of the bodies connected by the joint. For example, when two bodies are connected via a **prismatic joint**, only translation along some axis is permitted between the two bodies. In the case of a **revolute joint**, only rotation about some axis is allowed."},{"_id":"4dcd6c1a289482c4ce000059","treeId":"4dcbf502289482c4ce00004f","seq":1346610,"position":0.875,"parentId":null,"content":"The important thing to note is that the only kind of relative motion that joints permit is **rigid motion**. A **rigid transformation** of $latex \\mathbb{R}^n$\nis a map $latex x \\mapsto Ax + d$ for some orthogonal map $latex A$ with $latex det A > 0$ and some $latex d \\in \\mathbb{R}^n$.\n\nIt's seems helpful to note one thing: the class of all **isometries** on $latex \\mathbb{R}^n$ (i.e. all the maps $latex f: \\mathbb{R}^n \\to \\mathbb{R}^n$ such that $latex |f(x) - f(y)| = |x - y|$ for all $latex x, y \\in \\mathbb{R}^n$) is a strict superset of the class of rigid transformations."},{"_id":"4dcdc658289482c4ce00005e","treeId":"4dcbf502289482c4ce00004f","seq":1345855,"position":1,"parentId":"4dcd6c1a289482c4ce000059","content":""},{"_id":"4dcdc770289482c4ce00005f","treeId":"4dcbf502289482c4ce00004f","seq":1346611,"position":0.9375,"parentId":null,"content":"It can be proved that every isometry on $latex \\mathbb{R}^n$ is necessarily affine. So we proceed by studying the collection of affine isometries."},{"_id":"4dcbf632289482c4ce000052","treeId":"4dcbf502289482c4ce00004f","seq":1344785,"position":1,"parentId":null,"content":"The collection of affine isometries is exactly the set of all compositions $latex t \\circ k$ where $latex t$ is a translation and $latex k$ is a central isometry. *Proof:* (...)"},{"_id":"4dcbf8ee289482c4ce000053","treeId":"4dcbf502289482c4ce00004f","seq":1344786,"position":1,"parentId":"4dcbf632289482c4ce000052","content":"An **affine isometry** is an affine transformation which is also an isometry.\n\nA **translation** is any displacement of $latex \\mathbb{R}^n$ by a single fixed vector, for example $latex x \\mapsto x + d$.\n\nA **central isometry** is any isometry of $latex \\mathbb{R}^n$ that fixes zero. That is, an $latex f: \\mathbb{R}^n \\to \\mathbb{R}^n$ such that $latex f(0) = 0$."},{"_id":"4dcbfa26289482c4ce000054","treeId":"4dcbf502289482c4ce00004f","seq":1375222,"position":2,"parentId":"4dcbf632289482c4ce000052","content":"The collection of affine isometries on $latex \\mathbb{R}^n$ is exactly the collection of compositions $latex t \\circ k$ where $latex t$ is a translation of $latex \\mathbb{R}^n$ and $latex k$ is a central isometry. \n\n*Proof:* blah blah blah math goes here. $latex \\Box$"},{"_id":"4dcd2c5d289482c4ce000057","treeId":"4dcbf502289482c4ce00004f","seq":1345444,"position":2,"parentId":null,"content":"But one way to characterize the collection of all central isometries on $latex \\mathbb{R}^n$ is as the collection of general orthogonal maps on $latex \\mathbb{R}^n$."},{"_id":"4dcd2db4289482c4ce000058","treeId":"4dcbf502289482c4ce00004f","seq":1345447,"position":1,"parentId":"4dcd2c5d289482c4ce000057","content":"A **general orthogonal map** is a not necessarily linear function $latex f: \\mathbb{R}^n \\to \\mathbb{R}^n$ such that $latex f(x) \\cdot f(y) = x \\cdot y$ for all $latex x, y \\in \\mathbb{R}^n$."},{"_id":"4dcdbcbc289482c4ce00005b","treeId":"4dcbf502289482c4ce00004f","seq":1345852,"position":3,"parentId":null,"content":"It can be shown that every general orthogonal map is necessarily linear."},{"_id":"4dcdbdd1289482c4ce00005c","treeId":"4dcbf502289482c4ce00004f","seq":1345853,"position":4,"parentId":null,"content":"Hence the central isometries of $latex \\mathbb{R}^n$ are exactly all the elements of $latex O(n)$, the group of linear orthogonal maps, also known as the **orthogonal group**."},{"_id":"4dcdc176289482c4ce00005d","treeId":"4dcbf502289482c4ce00004f","seq":1345854,"position":1,"parentId":"4dcdbdd1289482c4ce00005c","content":"If $latex V$ is any vector space, the group $latex GL(V)$ of all bijective linear maps $latex V \\to V$ is called the **general linear group on $latex V$**. In the case of $latex V = \\mathbb{R}^n$, we write $latex GL(n)$ instead.\n\n$latex O(n)$ can be defined to be\n\n$latex O(n) := \\{ f \\in GL(n) : \\forall x, y \\in \\mathbb{R}^n, x \\cdot y = f(x) \\cdot f(y)$."},{"_id":"4dcdccc5289482c4ce000060","treeId":"4dcbf502289482c4ce00004f","seq":1346651,"position":5,"parentId":null,"content":"So not every isometry is a rigid transformation (but every rigid transformation is an isometry).\n\nNote that an isometry is an orthogonal map followed by a translation, however a rigid transformation is a rotation followed by a translation. So the rigid motions are what you get when you start with the isometries and throw out reflections (and glide reflections). throwing out reflections, since they do not preserve orientation."},{"_id":"4dd2f1f534d9adb37900001e","treeId":"4dcbf502289482c4ce00004f","seq":1346629,"position":1,"parentId":"4dcdccc5289482c4ce000060","content":""},{"_id":"4dd2f21634d9adb37900001f","treeId":"4dcbf502289482c4ce00004f","seq":1346630,"position":2,"parentId":"4dcdccc5289482c4ce000060","content":""},{"_id":"4dd346ed34d9adb379000020","treeId":"4dcbf502289482c4ce00004f","seq":1346652,"position":3,"parentId":"4dcdccc5289482c4ce000060","content":"A **glide reflection** is the composition of a translation and a reflection."}],"tree":{"_id":"4dcbf502289482c4ce00004f","name":"Rigid body motion","publicUrl":"rigid-body-motion"}}