## Excel AppNote: Generating Smooth Curves in Charts (WE0820)

The information in this article applies to:

- Microsoft Excel for Windows, version 4.0

## 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.

## MORE INFORMATION

### The Text of We0820

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    ======================================================================
      Microsoft(R) Product Support Services Application Note (Text File)
                  WE0820: GENERATING SMOOTH CURVES IN CHARTS
    ======================================================================
                                                       Revision Date: 8/93
                                                           1 Disk Included

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

    ----------------------------------------------------------------------
    | INFORMATION PROVIDED IN THIS DOCUMENT AND ANY SOFTWARE THAT MAY     |
    | ACCOMPANY THIS DOCUMENT (collectively referred to as an Application |
    | Note) IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER      |
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    | WARRANTIES OF MERCHANTABILITY AND/OR FITNESS FOR A PARTICULAR       |
    | PURPOSE. The user assumes the entire risk as to the accuracy and    |
    | the use of this Application Note. This Application Note may be      |
    | copied and distributed subject to the following conditions:  1) All |
    | text must be copied without modification and all pages must be      |
    | included;  2) If software is included, all files on the disk(s)     |
    | must be copied without modification (the MS-DOS(R)  utility         |
    | diskcopy is appropriate for this purpose);  3) All components of    |
    | this Application Note must be distributed together;  and  4) This   |
    | Application Note may not be distributed for profit.                 |
    |                                                                     |
    | Copyright (C) 1993 Microsoft Corporation.  All Rights Reserved.     |
    | Microsoft and MS-DOS are registered trademarks and Windows is a     |
    | trademark of Microsoft Corporation.                                 |
    | ------------------------------------------------------------------- |

    OVERVIEW
    ========

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

    INTRODUCTION
    ============

    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
    ===============================

    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
    ============================================================

    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:

    <chart deleted>
    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:

    <chart deleted>
    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
    ===========================================================

    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.)

    <chart deleted>
    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).

    <chart deleted>
    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 <b>, or
    to the right of <e>, then the parabolic equation that passes through
    these points is used (See Figure 7).

    <chart deleted>
    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).

    <chart deleted>
    Figure 8-Blending Curves 1 and 2

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

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

    <chart deleted>
    Figure 9-The Resulting Smooth Curve

    REFERENCES
    ==========

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

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## REFERENCES

Burger, P., and Gilles, D. "Interactive Computer Graphics, Functional, Procedural and Device Level Methods," 276-277. Palo Alto: Addison Wesley, 1992.
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Last Reviewed: December 10, 1999 © 2000 Microsoft Corporation. All rights reserved. Terms of Use.