4 Arithmetic of Segments—Hilbert`s Road from Ge
... Both EB AQ and EB AP , and both P and Q lie on the vertical line v. The uniqueness of the parallels and the uniqueness of intersection point implies P = Q . Hence ab = |OP | = |OQ | = ba, as to be shown. Remark. Conversely, the commutative law of multiplication, together with the ...
... Both EB AQ and EB AP , and both P and Q lie on the vertical line v. The uniqueness of the parallels and the uniqueness of intersection point implies P = Q . Hence ab = |OP | = |OQ | = ba, as to be shown. Remark. Conversely, the commutative law of multiplication, together with the ...
Quadrilaterals
... Proving that ABCD is a parallelogram If ABCD is a quadrilateral, then how can we prove that it is also a parallelogram? We must prove that both pairs of opposite sides are parallel. How can we prove that lines are parallel in a quadrilateral? To prove that lines are parallel, we must prove one of t ...
... Proving that ABCD is a parallelogram If ABCD is a quadrilateral, then how can we prove that it is also a parallelogram? We must prove that both pairs of opposite sides are parallel. How can we prove that lines are parallel in a quadrilateral? To prove that lines are parallel, we must prove one of t ...
Completed Notes
... Write down the angle measures, starting with 0° and continue until you reach 90°. Under these, write down the equivalent radian measures. Under these, write down the numbers from 0 to 4. Next, take the square root of the values and simplify if possible. Divide each value by 2. This gives you the sin ...
... Write down the angle measures, starting with 0° and continue until you reach 90°. Under these, write down the equivalent radian measures. Under these, write down the numbers from 0 to 4. Next, take the square root of the values and simplify if possible. Divide each value by 2. This gives you the sin ...
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.