Drawing Surfaces in MatLab

Transcription

1 Drawing Surfaces in MatLab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 7, 213 Outline Functions of Two Variables

2 Let s start by looking at the x y plane as a collection of two dimensional vectors. Each vector is rooted at the origin and the head of the vector corresponds to our usual coordinate pair (x, y). The set of all such x and y determines the x y plane which we will also call R 2. The superscript two is used because we are now explicitly acknowledging that we can think of these ordered pairs as vectors also with just a slight identification on our part. Since we know about vectors, note if we have a vector we can rewrite it, using our standard rules for vector arithmetic and scaling of vectors as [ 6 7 ] [ ] 1 [ ] = A little thought will let you see we can do this for any vector and so we define special vectors i = e 1 and j = e 2 as follows: [ ] [ ] 1 i = e 1 = and j = e 2 = 1 Thus, any vector can be written as [ ] x y = x e 1 + y e 1 = x i + y j Now let s start looking at functions that map each ordered pair (x, y) into a number. Let s begin with an example. Consider the function f (x, y) = x 2 + y 2 defined for all x and y.

3 Hence, for each x and y we pick, we calculate a number we can denote by z whose value is f (x, y) = x 2 + y 2. Using the same ideas we just used for the x y plane, we see the set of all such triples (x, y, z) = (x, y, x 2 + y 2 ) defines a surface in R 3 which is the collection of all ordered triples (x, y, z). Each of these triples can be identified with a three dimensional vector whose tail is the origin and whose head is the triple (x, y, z). We note any three dimensional vector can be written as x y = x e 1 + y e 2 + yz e 3 z = x i + y j + z k where we define the special vectors used in this representation by 1 i = e 1 =, j = e 2 = 1 and k = e 3 = 1 We can plot this surface in MatLab with fairly simple code. Let s go through how to do these plots in a lot of detail so we can see how to apply this kind of code in other situations. To draw a portion of a surface, we pick a rectangle of x and y values. To make it simple, we will choose a point (x, y ) as the center of our rectangle and then for a chosen x and y and integers n x and n y, we set up the rectangle [x n x x,..., x,..., x + n x x] [y n y y,..., y,..., y + n y y]

4 The constant x and y lines determined by this grid result in a matrix of intersections with entries (x i, y j) for appropriate indices i and j. We will approximate the surface by plotting the triples (x i, y j, z ij = f (x i, y j)) and then drawing a top for each rectangle. Right now though, let s just draw this base grid. In MatLab, first setup the function we want to look at. We will choose a very simple one f x, y ) x. ˆ 2 + y. ˆ 2 ; Now, we draw the grid by using the function DrawGrid(f,delx,nx,dely,ny,x,y). This function has several arguments as you can see and we explain them in the listing below. So we are drawing a grid centered around (.5,.5) using a uniform.5 step in both directions. The grid is drawn at z =. % Taking t h e arguments i n o r d e r % f i s the surface function % delx =. 5 i s the width of the delta x % nx = 2 i s the number of steps we take r i g h t and l e f t from % t h e b ase p o i n t x % dely =. 5 i s the width of the delta y % ny = 2 i s the number of steps we take r i g h t and l e f t from % t h e b ase p o i n t y % x =. 5 % y =. 5 DrawGrid ( f,. 5, 2,. 5, 2,. 5,. 5 ) ;

5 Make sure you play with the plot a bit. You can grab it and rotate it as you see fit to make sure you see all the detail. Right now, there is not much to see in the grid, but later when we plot the surface, the grid and other things, the ability to rotate in 3D is important to our understanding. So make sure you take the time to see how to do this! To draw the surface, we find the pairs (xi, yj) and the associated f (xi, yj) values and then call the DrawMesh(f,delx,nx,dely,ny,x,y) command. The meaning in the arguments is the same as in DrawGrid so we won t repeat them here. DrawMesh ( f,. 5, 2,. 5, 2,. 5,. 5 ) ;

6 The resulting surface and grid is shown here. Next, we draw the traces corresponding to the values x and y. The x trace is the function f (x, y) which is a function of the two variables y and z. The y trace is the function f (x, y ) which is a function of the two variables x and z. We plot these curves using the function DrawTraces. DrawTraces ( f,. 5, 2,. 5, 2,. 5,. 5 ) ;

7 The resulting surface with grid and traces is shown here. The traces are the thick parabolas on the surface. Next, let s add the column with the rectangular base having coordinates Lower Left (x, y ), Lower Right (x + x, y ), Upper Left (x, y + y) and Upper Right (x + x, y + y). We draw and fill this base with the function DrawBase. We draw the vertical lines going from each of the four corners of the base to the surface with the code DrawColumn and we draw and fill the patch of surface this column creates in the full surface with the function DrawPatch.

8 First, we draw the base. DrawBase ( f,. 5, 2,. 5, 2,. 5,. 5 ) ; In DrawColumn, we draw four vertical lines from the base up to the surface. You ll note the figure is getting more crowded looking though. Make sure you grab the picture and rotate it around so you can see everything from different perspectives. DrawColumn ( f,. 5, 2,. 5, 2,. 5,. 5 ) ; We then draw the patch just like we drew the base. DrawPatch ( f,. 5, 2,. 5, 2,. 5,. 5 ) ; The resulting surface with grid and traces and with the column and patch is shown in the next figure.

9 We have combined all of these functions into a utility function DrawSimpleSurface which manages these different graphing choices using boolean variables like DoGrid to turn a graph on or off. If the boolean variable DoGrid is set to one, the grid is drawn. The code is self-explanatory so we just lay it out here. We haven t shown all the code for the individual drawing functions, but we think you ll find it interesting to see how we manage the pieces in this one piece of code. So check this out.

10 First, let s explain the arguments f u n c t i o n D r a w S i m p l e S u r f a c e ( f, d e l x, nx, d e l y, ny, x, y, domesh, d o t r a c e s, d o g r i d, dopatch, docolumn, dobase ) % f i s t h e f u n c t i o n d e f i n i n g t h e s u r f a c e % d e l x i s t h e s i z e o f t h e x s t e p % nx is the number of steps l e f t and right from x 5 % d e l y i s t h e s i z e o f t h e y s t e p % ny is the number of steps l e f t and right from y % ( x, y ) i s the l o c a t i o n of the column r e c t a n g l e base % domesh = 1 means do the mesh, d o g r i d = 1 imeans do the g r i d % dopatch = 1 means add t h e p a t c h above t h e column 1 % dobase = 1 means add t h e base o f t h e column % docolumn = 1 add the column, dotraces = 1 add the t r a c e s % % s t a r t h o l d h o l d on Now look at the code 1 i f d o t r a c e s==1 % s e t up x t r a c e f o r x, y t r a c e f o r y DrawTraces ( f, d e l x, nx, d e l y, ny, x, y ) ; i f domesh==1 % p l o t t h e s u r f a c e 6 DrawMesh ( f, d e l x, nx, d e l y, ny, x, y ) ; i f dogrid==1 %plot x, y grid DrawGrid ( f, d e l x, nx, d e l y, ny, x, y ) ; 11 i f dopatch==1 % draw p a t c h f o r top o f column DrawPatch ( f, d e l x, nx, d e l y, ny, x, y ) ; i f dobase==1 16 % draw p a t c h f o r top o f column DrawBase ( f, d e l x, nx, d e l y, ny, x, y ) ; i f docolumn==1 %draw column DrawColumn ( f, d e l x, nx, d e l y, ny, x, y ) ; 21 h o l d o f f

11 Hence, to draw everything for this surface, we would use the session: >> f x, y ) x.ˆ2+ y. ˆ 2 ; >> D r a w S i m p l e S u r f a c e ( f,. 5, 2,. 5, 2,. 5,. 5, 1, 1, 1, 1, 1, 1 ) ; This surface has circular cross sections for different positive values of z and it is called a circular paraboloid. If you used f (x, y) = 4x 2 + 3y 2, the cross sections for positive z would be ellipses and we would call the surface an elliptical paraboloid. Now this code is not perfect. However, as an exploratory tool it is not bad! Now it is time for you to play with it a bit in the exercises below. Homework Explore the surface graph of the circular paraboloid f (x, y) = x 2 + y 2 for different values of (x, y ) and x and y. Experiment with the 3D rotated view to make sure you see everything of interest Explore the surface graph of the elliptical paraboloid f (x, y) = 2x 2 + y 2 for different values of (x, y ) and x and y. Experiment with the 3D rotated view to make sure you see everything of interest Explore the surface graph of the elliptical paraboloid f (x, y) = 2x 2 + 3y 2 for different values of (x, y ) and x and y. Experiment with the 3D rotated view to make sure you see everything of interest.