Triangle Angle Sum Theorem Proof
... Definition of a straight angle/Protractor Postulate Definition of transversal Definition of Alternate Interior Angles By construction If parallel, transveral, AIA, then congruent ...
... Definition of a straight angle/Protractor Postulate Definition of transversal Definition of Alternate Interior Angles By construction If parallel, transveral, AIA, then congruent ...
Trigonometric Ratios of Any Angle
... Rotation Angles Our work in trigonometry has so far been limited to angles between 00 and 900. If an angles has its vertex at the origin in the XY plane and its initial arm along the positive X-axis, then the measure of an angle can be extended beyond 900. We do this by rotating the terminal arm of ...
... Rotation Angles Our work in trigonometry has so far been limited to angles between 00 and 900. If an angles has its vertex at the origin in the XY plane and its initial arm along the positive X-axis, then the measure of an angle can be extended beyond 900. We do this by rotating the terminal arm of ...
Geometry Vocabulary #2
... 1) Ray - part of a line that begins at 1 point & extends without end in one direction. - named by its endpoint & one other point on it. Example: ...
... 1) Ray - part of a line that begins at 1 point & extends without end in one direction. - named by its endpoint & one other point on it. Example: ...
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.