Non –Euclidean Geometry
... Alternate Interior Angles Theorem Thm 4.1: Alterenate Interior Angles (AIA): If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines parallel. Corr 1: Two lines perpendicular to the same line are parallel. Corr 2 (Euc I.31): If l is an line and P is ...
... Alternate Interior Angles Theorem Thm 4.1: Alterenate Interior Angles (AIA): If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines parallel. Corr 1: Two lines perpendicular to the same line are parallel. Corr 2 (Euc I.31): If l is an line and P is ...
Lesson 2: Bisect and Copy an Angle
... Before we do this, we need to clarify our assumptions about measuring angles. These assumptions are based upon what we know about a protractor that measures up to 180˚ angles: ...
... Before we do this, we need to clarify our assumptions about measuring angles. These assumptions are based upon what we know about a protractor that measures up to 180˚ angles: ...
Quarter 2
... G-SRT.A.01 Verify experimentally the properties of dilations given by a center and a scale factor: G-SRT.A.01.a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.A.01.b The dilation of a line segme ...
... G-SRT.A.01 Verify experimentally the properties of dilations given by a center and a scale factor: G-SRT.A.01.a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.A.01.b The dilation of a line segme ...
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.