Lesson F5.3

... You have learned how to identify different types of polygons, including triangles, and how to find their perimeter and their area. In this lesson, you will learn more about triangles. You will study the properties of triangles in which two or three lengths or two or three angles have the same measur ...

... You have learned how to identify different types of polygons, including triangles, and how to find their perimeter and their area. In this lesson, you will learn more about triangles. You will study the properties of triangles in which two or three lengths or two or three angles have the same measur ...

Summary of Objectives

... the end of a transformation. Part II Discuss what a tessellation is and what figure will and will not tessellate. ...

... the end of a transformation. Part II Discuss what a tessellation is and what figure will and will not tessellate. ...

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or

... 17. ROADWAYS Main Street and High Street intersect as shown in the diagram. Each of the crosswalks is the same length. Classify the quadrilateral formed by the crosswalks. Explain your reasoning. ...

... 17. ROADWAYS Main Street and High Street intersect as shown in the diagram. Each of the crosswalks is the same length. Classify the quadrilateral formed by the crosswalks. Explain your reasoning. ...

HOW STABLE AM I? TRIANGLE CONGRUENCE

... Let’s begin this lesson by finding out what congruent triangles are. As you go over the activities, keep this question in mind. “When are two triangles congruent?” Activating Prior Knowledge ...

... Let’s begin this lesson by finding out what congruent triangles are. As you go over the activities, keep this question in mind. “When are two triangles congruent?” Activating Prior Knowledge ...

# Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.