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A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined.
1 Definition of a Surface. There are many definitions of a surface, and they all turn out to be equivalent, provided you define the notion of equivalence correctly. I’ll give a definition which is well-adapted to drawing graphs using polygonal edges.
Surfaces as graphs of functions. If f f is a scalar-valued function of a single variable, f:R →R f: R → R (confused?), then the graph of f f is the set of points (x, f(x)) (x, f (x)) for all x x in the domain of f f. When often call this the graph of y = f(x) y = f (x), since we think of the points as lying in the xy x y -plane.
surf(X,Y,Z) creates a three-dimensional surface plot, which is a three-dimensional surface that has solid edge colors and solid face colors. The function plots the values in matrix Z as heights above a grid in the x - y plane defined by X and Y .
Graphs: For g(x,y,z) = z − f(x,y) we have the level surface g = 0 which is the graph z = f(x,y) of a function of two variables. For example, for g(x,y,z) = z−x2 −y2 = 0, we have the graph z = x2 + y2 of the function f(x,y) = x2 + y2 which is a paraboloid. Note however that most surfaces of the form g(x,y,z) = c can not be written as graphs.
A surface is the graph of a nice function. Example 1. The graph of f(x; y) = x2 + y2 de nes a paraboloid. Example 2. It is awkward to de ne the unit sphere this way.
Surface Chart (3D Surface Plot) displays a set of three-dimensional data as a mesh surface. It is useful when you need to find the optimum combinations between two sets of data. The colors and patterns in Surface Charts indicate the areas that are in the same range of values by analogy with a topographic map.
