![]() ![]() Through the establishment of a solid foundation of precise vocabulary and developing arguments in Unit 1, students are able to use these strategies and theorems to identify and describe geometric relationships throughout the rest of the year. In the next unit, students use the concepts of constructions, proof, and rigid motions to establish congruence with two dimensional figures. Students focus on rigid motions with points, line segments and angles in this unit through transformation both on and off the coordinate plane. Students learn that rigid motions can be used as a tool to show congruence. Topic C merges the concepts of specificity of definitions, constructions, and proof to formalize rigid motions studied in 8th Grade Math. In Topic B, students formalize understanding developed in middle school geometry of angles around a point, vertical angles, complementary angles, and supplementary angles through organizing statements and reasons for why relationships to construct a viable argument. Students are introduced to the concept of a construction, and use the properties of circles to construct basic geometric figures. ![]() ![]() Unit 1 begins with students identifying important components to define- emphasizing precision of language and notation as well as appropriate use of tools to represent geometric figures. This unit lays the groundwork for constructing mathematical arguments through proof and use of precise mathematical vocabulary to express relationships. These rigid motion transformations are introduced through points and line segments in this unit, and provide the foundation for rigid motion and congruence of two-dimensional figures in Unit 2. Transformations that preserve angle measure and distance are verified through constructions and practiced on and off the coordinate plane. Note that p^0 and p^1 point to the same location although p^0 \neq p^1 due to the choice of different coordinate frames.In Unit 1, students are introduced to the concept that figures can be created by just using a compass and straightedge using the properties of circles, and by doing so, properties of these figures are revealed. The following script specifies the coordinates of a point p with respect to two different coordinate frames o_0 x_0 y_0 and o_1x_1y_1 on the two-dimensional space. Mathematically, the position of a point p can be represented as n-tuple such that p \in \mathbb^n where n = 2 or 3. A coordinate frame consists of an origin (a single point in space), and two or three orthogonal coordinate axes, for two- and three-dimensional spaces, respectively.Ī point is a geometric entity which corresponds to a specific location in the space. The mathematical representation of those entities requires the choice of a reference coordinate frame. ![]() Furthermore, representing the forces and the torques is essential to analyze and design robots. To accomplish certain tasks, engineers need to represent the positions and the velocities of those special points. Representing Positions and VectorsĪ robot has several special points on its body such as the center of mass, the tip of its end effector, etc. Finally, it presents homogeneous transformations. This chapter introduces mathematical representations of positions and vectors, rotations and displacements. Understanding rigid motions allow robot programmers to describe the position and orientation of the end effector of a robot. A homogeneous transformation matrix combines a rotation matrix with a displacement vector to represent those two properties simultaneously. Therefore, there are two essential kinematic properties of a rigid body: (i) orientation, (ii) position. Owing to this assumption, a rigid body motion is a combination of rotation and translation. Consequently, the distance between any two points on a rigid body is assumed to remain constant in time during any motion. A rigid body is a solid body in which deformation is negligibly small. ![]()
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