PH36010 MathCAD worksheet Advanced Graphics and Animation

Transcription

1 PH361 MathCAD worksheet Advanced Graphics and Animation In this worksheet we examine some more of mathcad's graphing capabilities and use animation to illustrate aspects of physics. Polar Plots A polar plot is simply an x-y graph wrapped around into a circle. After the wrapping, the x-axis represents the angles between and 36 deg and the y-axis the radius. To illustrate a simple polar plot, we will examine the curve known as the Cardoid. This is a common curve that appears in the directional response of microphones and antennae. The Cardoid curve gives a radius, r, and a function of an angle, θ. The equation of the curve is given by: r(θ)=a[cos(θ)+1] where a is a constant, in this case we will take a=1 First define the function r(θ) a := 1 := a ( cos( θ) + 1) r θ Now call up a Polar plot (Insert Graph Polar Plot) and fill in the placeholders for θ and r Note: MathCAD automatically creates a range for θ going from to 2π (-36) r( θ) Although mathcad is working in radians, the polar plot is scaled in degrees. This is a slight inconsistency within mathcad, but don't worry θ (C) dpl 22 1/18

2 As well as displaying continuous functions of angle like we did above, polar plots can also display discrete points of data stored in vectors, in the same way as x-y plots. In the photon scattering example, experimental data of scattering angles was given in a text file containing 2 columns of values. Data :=...\phase.csv Angle Data deg Read data into matrix NB. If you are copying this worksheet into your own directory, you will need to copy the data file and change the file read component to point to it. Follow link from users.aber.ac.uk/dpl -> Course Material -> PH361 -> Phase.csv := First column contains angles in degrees LogPhase Data 1 := Second column contains log1 of response 8 6 X Y plot of data LogPhase Angle We can plot the same data as a polar plot LogPhase Angle 3 (C) dpl 22 2/18

3 Change the trace to stem in order to plot lines from the origin to each data point. LogPhase Angle 3 Note how the data only gives the response for half the circle, which is reflected in the plot. It is assumed that the response in the range is the same as that in the -18 range. Using matrix functions from the f(x) toolbar button, create two vectors, LogPhase36 and Angle36 which represent the data for the angles -18 to 18. Create a plot showing the complete polar scattering function. Hints: Create the Angle36 vector first, then you can easily check you have a sensible looking answer before using similar steps to create the LogPhase36. Angle contains the anlges between and 18, so the reverse of Angle will contain the angles between 18 and. You can stack two vectors together to make a single long one. You need to make sure that you don't get two zeroes in the middle of your result vector. How would you use submatrix() to return a vector missing its first row? If V is a vector, then what is -V? Functions that may be useful are: reverse(), stack(), submatrix() (C) dpl 22 3/18

4 Simple 3D surface plots Here we will use a simple 3D surface plot to show the profile of a gaussian surface. Gaussian surfaces arising as a result of many natural processes, for example if we shine a narrow beam of ions or photons onto a flat plate and measure the intensity of the beam at different places on the plate, we will find the result usually looks like a gaussian. The gaussian functions we will plot is a function of x- and y- and looks like: k := 1 Gaussian2D( x, y) := e ( x 2 + y 2 ) k k is a constant which determine the width of our beam. In plotting, we frequently need to fill a vector with a range of values. This makes it easy to scale our plots as required. Since we will be doing this several times in this sheet, it makes sense to define a program to do it for us... Max Min FillSeries( Min, Max, N) N := Calculate the increment for i.. N Result i Min + i Result Loop through the number of increments, adding each one to the result When finished looping, return the result Having created the program, test it to check it works before moving on. FillSeries( 1, 1, 1) = Returns a vector going from -1 to 1 in 1 steps. Notice that in order to take 1 steps, we visit 11 points Now the FillSeries() functions appears to be OK, we can move on... (C) dpl 22 4/18

5 We will plot our Gaussian2D() function over x and y values going from -5 to +5 in 1 steps. Create variable to hold the minimum and maximum values for x and y and also the number of steps in each direction. N := 1 xmin := 5 xmax := 5 ymin := 5 ymax := 5 Use the FillSeries() function to create vectors X and Y which hold the x and y values we wish to plot the function at X := FillSeries( xmin, xmax, N) Y := FillSeries( ymin, ymax, N) Use the space on the right of the page to print out the 2 vectors and assure yourselves that they look OK. We now wish to fill a matrix, Z, with the values of the Gaussian2D function at the corresponding point. To do this we create 2 range variables, i & j, which we use to step through the X & Y vectors i :=.. N i is used to step the the X vector j :=.. N j is used to step the the Y vector Filling the Z matrix is done with the range variables i & j Z i, j := Gaussian2D X i, Y j Now insert a 3D surface plot and display the Z-matrix This is the default plot format. The surface is not filled in and the lines are all black Bring up the format box and on the Appearance tab page, change the surface to be filled with a colourmap and just contour lines to be drawn. Z (C) dpl 22 5/18

6 Experiment with different options on the pages on the format dialog box Try dragging the mouse over the plot to get different view angles. Hold <shift> whilst dragging the mouse to start the plot rotating. Z Experiment with changing the values above for xmin,xmax,ymin and ymax as well as the k constant. (C) dpl 22 6/18

7 The following segment of mathcad will plot a cone a := 2 Base radius b := 5 Height of cone h( x, y) b := a x 2 + y 2 cone( x, y) := if x 2 + y 2 + b a 2 >,, h( x, y) Function h(x,y) gives height of slope Function cone(x,y) uses a conditional to restrict the points on the cone to those above the xy plane Z i, j cone X i, Y j := Fill the Z matrix with the cone() function, using the same range of X & Y values as before. Z Use the plot format options to display the cone with a colourmap over the surface. On the advanced tab-page of the format dialog box, you can select different colour maps to use. A truncated cone is simply a cone with the top sliced off. Create a function TruncCone(x,y) that uses the definitions of h(x,y) and cone(x,y) togther with an if() function to give a cone with the top sliced off at a height of zz. Plot your truncated cone with zz set to 3 Try and create a function to plot a hemisphere. You may need to adjust the scaling on the z-axis so that it looks like a hemisphere. (C) dpl 22 7/18

8 Advanced 3D surface plots So far the 3D surface plots have lacked any scaling in the x and y axes. The numbers appearing on the plot axes have related to the indices of the z matrix rahter than 'real world' co-ordinates. In this section we will combine 3 matrices together in order to give plots where the X and Y axes are scaled in real world co-ordinates. We will use the X and Y vectors we created earlier, together with the range variables for indexing them. XX i, j := X i YY i, j := Y j XX = YY = ZZ i, j := Gaussian2D XX i, j, YY i, j The XX and YY matrices hold the real world co-ordinates of the points we wish to plot. Note how the indices are used. Here is a section of the XX and YY matrices. See how each column of the XX matrix is identical and each row of the YY matrix is identical. When we fill the Z matrix using the range variables, the Gaussian2D function is called with the x and y values from the corresponding point in the XX and YY tables. Thus Z, becomes Gaussian2D(-5,-5) Z,1 becomes Gaussian2D(-5,-4.9) Z,2 becomes Gaussian2D(-5,-4.8)... Z 1, becomes Gaussian2D(-4.9,-5)... Z 1,99 becomes Gaussian2D(5,4.9) Z becomes Gaussian2D(5,5) (C) dpl 22 8/18

9 Z 1,1 becomes Gaussian2D(5,5) Once we have filled the matrices we can plot the function by putting the 3 matrices in the 3D plot placeholder, seperated with commas and surrounded in brackets. ( XX, YY, ZZ) Notice how the axes are now scaled in real numbers over the range we specified earlier. Experiment with the shading and colourmap options as before (C) dpl 22 9/18

10 Polar plots in 3D In the above example, we used a simple scaling relationship to create the XX and YY matrices. We can use more complicated relationships to make plots and surfaces to fit in with our problems. In this example we will map our XP and YP matrices onto polar co-ordinates, enabling us to plot functions which are essentially polar in nature. In this case we will plot the electron probability function for the 1 electron in the Hydrogen atom. The first step is to define our range of r and θ values we are going to be plotting If you are pusing a version of mathcad prior to v13, you will need to define nm as 1-9 m and a as m. a is the bohr radius of the atom RSteps := 3 We will plot 3 steps in the radial direction Create a vector Radius holding a series of radii from m to 6 Bohr radii. Radius := FillSeries m, 6 a, RSteps ri :=.. RSteps Range variable ri for indexing radii θsteps := 2 2 steps in angular direction Create vector Theta holding a series of angles from -π to π Theta := FillSeries π, π, θsteps θi :=.. θsteps Range variable θi for indexing angles Now, once we've created the Theta and Radius vectors, along with the range variables for indexing them, we can fill our XP and YP matrices with polar coordinates. XP Radius ri, θi ri cos Theta θi := NB If you try and re-use the XX or YY matrices, you will get a Units error - why? YP := Radius ri, θi ri sin Theta θi (C) dpl 22 1/18

11 We can examine the XP and YP matrices as before: XP = m YP = m In the XP and YP matrices, the angular values go across and the radius values go down the page. The XP matrix holds the x-coordinates of the points in the plane and the YP matrix holds the y- co-ordinates of the points. Thus, at an angle of -π radians (first column) all the x-coordinates are negative and the y co-ords zero. At an angle of -9 deg (column 5) the x-coordinates are zero and the y values negative. Having created our XP and YP matrices, we can see about plotting a function on our 3D polar graph. ψ1( r) := r 1 3 e a πa The wave function ψ of the 1 electron as a function of radius (C) dpl 22 11/18

12 We can start by plotting ψ1() in two dimensions. This is easily achived using the Radius vector we defined earlier, along with the ri range variable ψ1 Radius ri Radius ri Now we can fill a matrix with the result of applying our function 3 Z1 := πa ri, θi ψ1 Radiusri XP YP,, Z1 m m XP YP,, Z1 a a In the left hand plot we have divided the XP and YP matrices by m to scale the plot in meters In the right hand plot, we have divided the XP and YP matrices by a, in order to scale the axes in Bohr radii. The ψ1 and ψ2 functions are both symetrical about the axis and do not depend on θ. The ψ21 function depends on θ as well as r and is given below ψ21( r, θ) r 1 2 a r e cos θ 5 32 π a := Create a matrix Z21 and fill it with the ψ21 function over the range of values before plotting it. (C) dpl 22 12/18

13 Plotting a sphere and other surfaces The following snippet is taken from the resource centre and shows how to plot a sphere by the simple application of spherical co-ordinates. Note that I've changed the names of the matrices to avoid conflicting with the X,Y and Z matrices we used earlier in this sheet. Np := 25 π i :=.. Np φ i i Np π j :=.. Np θ j := j 2 Np := Could use FillSeries here. SphereX i, j := sin φ i SphereY i, j := sin φ i SphereZ i, j := cos φ i cos( θ j ) sin( θ j ) These are simply the equations to convert spherical coordinates into cartesian. Play with plot options ( SphereX, SphereY, SphereZ) (C) dpl 22 13/18

14 Having created a single sphere, it is trivial to plot two of them on the same graph. Here the second sphere is moved 2.1 units along the x-axis ( SphereX, SphereY, SphereZ), ( SphereX + 2.1, SphereY, SphereZ) Try selecting 'Equal Scales' from the plot formatting options to get a good display. You can select different fill options for each plot on the graph. The resource centre has quicksheets which act as a starting point for plotting spheres, tori and other more complex surfaces. (C) dpl 22 14/18

15 Animation in mathcad Animations consist of a number of pictures shown one after another in succession. this fools the eye into believing that it is observing continuous motion. MathCAD performs animations by means of a system variable called FRAME. When you ask mathcad to animate a region of the worksheet, it takes a snapshow of the region before incrementing FRAME and taking another snapshot. When all the snapshots are combined together they give the impression of continuous motion. As a trivial, if not particularly exciting example, bring up the animation dialog box (Tools Animation Record...) When the dialog box is displayed, use the mouse to drag a box around the region where the value of FRAME is displayed below. Once you have selected the region to animate, press the Animate button in the dialog box. After a while, you should see the windows media player and be able to play back a little animation showing how FRAME changes from to 9 FRAME = Colngratulations, you have created your first animation. More exciting animations usually contain graphs. Have a look at some of the examples in the Resource Centre. There is a complete tutorial guide in Advanced Topics Treasury Guide to Animations and a few very useful examples in Quicksheets Animations (C) dpl 22 15/18

16 The following example shows how simple it is to animate the spheres we created earlier. Try animating the video for FRAME going from to 2 π := Displacement( t) := 2 + sin t 1 t := FRAME Before you begin animation, select format the graph and fix the x axis scale to -4 to +4 ( SphereX Displacement( t), SphereY, SphereZ), ( SphereX + Displacement( t), SphereY, SphereZ) Experiment with lighting, appearance, surface plot and other options (C) dpl 22 16/18

17 Z1 = (C) dpl 22 17/18

18 (C) dpl 22 18/18