Microsoft KB Archive/103841

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Excel AppNote: Generating Smooth Curves in Charts (WE0820)

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The information in this article applies to:


 * Microsoft Excel for Windows, version 4.0

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SUMMARY
The Application Note "Generating Smooth Curves in Charts" (WE0820) includes two add-in macros that use the interpolation method to generate smooth curves in charts.

To Obtain This Application Note
The following file is available for download from the Microsoft Download Center. Click the file name below to download the file:

"We0820.exe" For more information about how to download files from the Microsoft Download Center, please visit the Download Center at the following Web address

"http://www.microsoft.com/downloads/search.asp" and then click How to use the Microsoft Download Center.

The Text of We0820


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========================================================= Microsoft(R) Product Support Services Application Note (Text File) WE0820: GENERATING SMOOTH CURVES IN CHARTS

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=========================================================                                                  Revision Date: 8/93 1 Disk Included

The following information applies to Microsoft Excel for Windows(TM), version 4.0.

--
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 * Copyright (C) 1993 Microsoft Corporation. All Rights Reserved.     |
 * Microsoft and MS-DOS are registered trademarks and Windows is a    |
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 * trademark of Microsoft Corporation.                                |

OVERVIEW

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This Application Note includes two add-in macros that use the interpolation method to generate smooth curves in charts.

INTRODUCTION

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In Microsoft Excel 4.0, when you create an xy (scatter) chart, the series of xy data points generally do not form a smooth curve. Using a method called interpolation, you can generate a series of values based on a set of xy data points that, when plotted, appear as a smooth curve that connects your xy data points. Interpolation achieves this smooth-curve effect by joining together several smaller curves called splines.

The enclosed WE0820 disk includes two add-in macros, INTERPOL.XLA and BLEND.XLA, that work together to generate these interpolated values based on x and y data ranges that you specify. The INTERPOL.XLA macro uses a custom dialog box to prompt you for your input range information and allows you to specify how you want your data to be returned. Based on your specifications, BLEND.XLA calculates and returns the data points that will generate a smooth curve through the given arrays of x and y values. You also have the option to create a chart that displays the new values in a curve and the original values as data points.

NOTE: When plotted, your x and y values can be in almost any form (for example, your values can form one or more loops or rotate through 90 degrees, and so on). In addition, if the original x and y values change, the data returned by BLEND.XLA will automatically be  updated.

The following information explains how to install and use these add-in macros and then discusses their theoretical basis.

INSTALLING AND USING THE MACROS

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To install the add-in macro files -

1. In File Manager, copy the BLEND.XLA and INTERPOL.XLA files from the enclosed WE0820 disk to the LIBRARY subdirectory on your hard disk (this subdirectory is located in the directory where you installed  Microsoft Excel).

2. In Microsoft Excel, choose Open from the File menu. Select the INTERPOL.XLA file and choose the OK button.

When you open the INTERPOL.XLA file, the BLEND.XLA file is automatically loaded and the Interpolate command is added to the Data menu.

To specify options in the Data Interpolate dialog box -

When you choose Interpolate from the Data menu, the Data Interpolate box appears, prompting you for certain information. In this dialog box, you are requested to enter information about your existing data, and you are given options for how you want to present the data that the macro generates. Once you have entered the appropriate information in the dialog box, the macro will return a two-column vertical array of x and y values. These values are the additional data points you'll need to make a smooth curve.

The following tables explain what you should enter in each box and discuss the specific options you can select to control how the new data is displayed.

In this box         Enter the range of   ---

Known X Values      Your known x values Known Y Values      Your known y values

NOTE: Your data must consist of at least four x values and four y   values; these values can be in rows or columns.

In this box         Enter the following

Output Reference    Specify the first cell in                        which the interpolated data will be pasted

NOTE: If you select a cell that contains data or is located above data that will be overwritten, you will receive an alert message. When you choose OK in this alert dialog box, you will be returned to the Data Interpolate dialog box to make another cell selection.

In the Number Of Intermediate Data Points box, you can specify the number of intermediate points you think you will need to generate a smooth curve--the initial value displayed for the number of intermediate points is calculated based on several factors, including screen resolution, printer resolution, and "smoothness" of the original data. In most cases, ten intermediate data points should be enough to generate a smooth curve.

In this box        Enter the following ---

Number Of          The number of additional Intermediate       data points between your Data Points        original data points that you will need to generate a smooth curve

To control how your data will be displayed, use the options in the Options section of the dialog box.

Select this option  To do this ---

Include Original    Include the xy values on   Points               which your interpolated data is based

Create Chart        Create a chart with a                        smooth curve

When you select both of these options, the add-in macros work together to create a chart with two data series: one for the smooth curve that has a line style but no data point markers and one for the original data that has data point markers but no line style.

NOTE: You can use the BLEND.XLA macro independent of INTERPOL.XLA to return data points along your smooth curve. You can use the Paste Function command on the Formula menu to paste this data to   your sheet. The function takes three arguments: the range containing your x values, the range containing your y values, and the number of intermediate points you want. To enter the formula, you must have at least two cells selected (either columns or rows) and you must enter it as an array formula by pressing CTRL+SHIFT+ENTER.

USING THE INTERPOL.XLA AND BLEND.XLA MACROS WITH SAMPLE DATA

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The following sample data

|  A   |  B - 1 |   1      4 2  |   2      2 3  |   3      2 4  |   4      4 5  |   5      3

Figure 1-Sample Data

when plotted in a chart will resemble the following:

Figure 2-Sample Data Plotted in a Chart

To generate a new xy (scatter) chart with a smooth curve that is based on the data in cells A1:B5, do the following:

1. From the Data menu, choose Interpolate. In the startup message box, choose OK.

2. In the dialog box that is displayed, do the following:

a. In the Known X Values box, type "$A$1:$A$5" (without the quotation     marks).

b. In the Known Y Values box, type "$B$1:$B$5" (without the quotation      marks).

c. In the Output Range box, type "$C$1" (without the quotation marks).

d. In the Number Of Intermediate Points box, leave the value set to      10.

e. Under Options, verify that Include Original Points and Create Chart are both selected.

3. Choose OK.

The following table shows the first 15 of 45 rows of the resulting data (the total number of rows will vary with the number of x and y values and the number of intermediate points that you specify).

|    C     |    D     |    E    --- 1 |         1          4      4    2  |  1.090909   3.735537    3  |  1.181818   3.487603    4  |  1.272727   3.256198    5  |  1.363636   3.041322    6  |  1.454545   2.842975    7  |  1.545455   2.661157    8  |  1.636364   2.495868    9  |  1.727273   2.347107   10  |  1.818182   2.214876   11  |  1.909091   2.099174   12  |         2          2      2   13  |  2.090909   1.917355   14  |  2.181818    1.85124   15  |  2.272727   1.801653

Figure 3-Table of Intermediate Data Point Values

Based on these values, INTERPOL.XLA automatically creates the following chart:

Figure 4-Interpolated Curve

In Figure 4, the line shows the interpolated curve, and the data point markers show the original xy values.

THEORETICAL BASIS FOR THIS METHOD OF CREATING SMOOTH CURVES

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This particular implementation of curve smoothing is defined as the construction of a smooth curve by the piecewise addition of linearly blended second order polynomials (Burger and Gilles 1992).

An explanation of this method is as follows: given any three non- singular points, it is possible to define a unique parabola that will pass through all three of them (Burger and Gilles 1992). (See Figure 5.)

Figure 5-A Unique Parabola that Passes Through Three Given Points

To generate a curve through several points, parabolas are calculated for each set of three points (see Figure 6).

Figure 6-The Unique Parabolas that Pass Through Each Set of Three Points

If the interpolated point we want to find lies to the left of , or to the right of , then the parabolic equation that passes through these points is used (See Figure 7).

Figure 7-The End Conditions

For each intermediate section of the curve, the two parabolas that pass through each point are added together in proportion to how close they are to the original points (see Figure 8).

Figure 8-Blending Curves 1 and 2

For instance, if the desired point lies between  and , then the closer the point is to , the more of curve 1 is used; and conversely, the closer the point is to , the more of curve 2 is used. At point , only curve 1 is used; at point , only curve 2 is used; and halfway between  and , the interpolated point is the average of curves 1 and 2.

Continuing in this way, you can construct a smooth curve (see Figure 9).

Figure 9-The Resulting Smooth Curve

REFERENCES

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Burger, P., and Gilles, D. "Interactive Computer Graphics, Functional, Procedural and Device Level Methods," 276-277. Palo Alto: Addison Wesley, 1992.