Cartesian plane in the context of Coordinate frame

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted ${\displaystyle {\textbf {E}}^{2}}$ or ${\displaystyle \mathbb {E} ^{2}}$. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.

A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.The set ${\displaystyle \mathbb {R} ^{2}}$ of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.

In mathematics, the upper half-plane, ⁠${\displaystyle {\mathcal {H}},}$⁠ is the set of points ⁠${\displaystyle (x,y)}$⁠ in the Cartesian plane with ⁠${\displaystyle y>0.}$⁠ The lower half-plane is the set of points ⁠${\displaystyle (x,y)}$⁠ with ⁠${\displaystyle y<0}$⁠ instead. Arbitrarily oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants.

In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. Hyperbolic angle is a shuffled form of natural logarithm as they both are defined as an area against hyperbola xy = 1, and they both are preserved by squeeze mappings since those mappings preserve area.

The hyperbola xy = 1 is rectangular with semi-major axis ${\displaystyle {\sqrt {2}}}$, analogous to the circular angle equaling the area of a circular sector in a circle with radius ${\displaystyle {\sqrt {2}}}$.

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups ${\displaystyle A}$ and ${\displaystyle B}$ is another abelian group ${\displaystyle A\oplus B}$ consisting of the ordered pairs ${\displaystyle (a,b)}$ where ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$. To add ordered pairs, the sum is defined ${\displaystyle (a,b)+(c,d)}$ to be ${\displaystyle (a+c,b+d)}$; in other words, addition is defined coordinate-wise. For example, the direct sum ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$, where ${\displaystyle \mathbb {R} }$ is real coordinate space, is the Cartesian plane, ${\displaystyle \mathbb {R} ^{2}}$. A similar process can be used to form the direct sum of two vector spaces or two modules.

Direct sums can also be formed with any finite number of summands; for example, ${\displaystyle A\oplus B\oplus C}$, provided ${\displaystyle A,B,}$ and ${\displaystyle C}$ are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). That relies on the fact that the direct sum is associative up to isomorphism. That is, ${\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}$ for any algebraic structures ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ of the same kind. The direct sum is also commutative up to isomorphism, i.e. ${\displaystyle A\oplus B\cong B\oplus A}$ for any algebraic structures ${\displaystyle A}$ and ${\displaystyle B}$ of the same kind.

A 2D geometric model is a geometric model of an object as a two-dimensional figure, usually on the Euclidean or Cartesian plane.

Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of sheet metal. Other examples include circles used as a model of thunderstorms, which can be considered flat when viewed from above.