Curves are represented in Cartesian Coordinate Systems using Explicit Scalar Equations, Implicit Scalar Equations and / or Parametric Position Vector Functions.
Explicit Scalar Equations: These equations can be used to represent Curves in 2D Cartesian Coordinate System.
The Curves are represented as
\(y=f(x)\)
where \(x\) is the Independent Variable and \(y\) which is a Function of Variable \(x\) is the Dependent Variable
OR
\(x=f(y)\)
where \(y\) is the Independent Variable and \(x\) which is a Function of Variable \(y\) is the Dependent Variable.
Please note the although Explicit Scalar Equations allow us to determine the Direction of the Curves, not all kinds of 2D Curves can be represented by using these equations, especially in which the Curves Intersect/Meet themselves.
Implicit Scalar Equations: These equations can also be used to represent Curves in 2D Cartesian Coordinate System.
The Curves are represented as
\(f(x,y)=0\)
Please note that Although any kind of 2D Curves can be represented by using the Implicit Scalar Equations, the Direction of the Curves cannot be determined by by these equations.
Parametric Position Vector Functions: These functions can be used to represent any and all kinds of Curves in 2D and 3D Cartesian Coordinate System.
Also representing Curves using these functions allow us to determine the Direction of the Curves.
The Curves are represented using Position Vectors whose Components are Functions of a Single Variable Parameter. For example, any 2D Curve is represented as
\(\vec{R}=x\hat{\mathbf{i}} + y\hat{\mathbf{j}}\)
where \(x\) and \(y\) are Functions of a Single Variable Parameter \(t\) given as follows
\(x=f_x(t)\), \(y=f_y(t)\)
and Vector \(\vec{R}\) is the Position Vector of any point on the Curve.