# Koman Surfaces

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Draw a curve on a balloon, connect the points on the curve to the center of the balloon with lines, and then pop the balloon. The resulting shapes can be very pleasing and artistic. Ilhan Koman, an artist from Istanbul, Turkey, has produced very elegant sculptures using this method. This Demonstration lets you create such sculptures and then rotate and zoom them in 3D.

Contributed by: Karl Scherer (May 2011)
Open content licensed under CC BY-NC-SA

## Details

The sculptures you can create with this Demonstrations consist of arcs (partial circles), lines, and spirals on the surface of a sphere. These curves are all connected with one central point, the center of the sphere.

Polar coordinates are used to position the centers of the arcs (via sliders), but no knowledge of mathematics is necessary to use this Demonstration.

An arc is defined by a central point, a radius, and two angles that measure the arc's start and end positions as an angle around the arc's center.

A spiral has two radii (distances to the spiral's center), one for the start position and one for the end position. Between the end points the radius smoothly changes from one value to the other.

The examples are provided to give you a good idea of how to start your own sculpture.

The currently edited arc or spiral can be automatically linked to another arc or spiral without gaps or overlaps by shifting it to the appropriate position. See details for "join to part" and "gr/gg/rr/rg" below.

The sculpture "9" is the only example given that uses spirals.

The controls are described in the following.

"sculpture" pop-up menu and selector ""

Use these controls to select the sculpture you would like to look at or edit.

The sculptures are sets of joined or unjoined curved pieces on a sphere. Such a piece can be an arc (a partial circle) or a partial spiral, where all points of the curved piece are connected with a common central point (the center of the unit sphere at the origin).

You can edit an existing sculpture or create your own from scratch.

Use this control to select the arc (or spiral) you want to edit.

The start and end points of this curve will be marked with a green and a red point. Also, the center of the part will be marked by a larger red point.

An arc or spiral is implemented as a series of straight lines. The vertices of the current part are marked by black points.

The surface of the sculpture consists of triangles defined by two consecutive vertices of the arc or spiral and the center of the sphere.

"</>>"

Click one of the buttons to jump to the first, previous, next, or last arc. Warning: always click "save" before you select another part; all editing of the current part will be lost otherwise.

Click "save" to save the current arc or spiral.

Click "restore" to restore the current arc or spiral.

Click "default" to restore the default of the current part. If there is no default because you have added this part, then the default of the first part of the sculpture will be shown as the default of this added part.

"no action" is the default value of a list of actions you can perform on your arc or spiral.

Select "delete" to delete the currently edited part. Note that the last existing part cannot be deleted.

Select "delete others" to delete all arcs and spirals except the current one. The last remaining part cannot be deleted.

Select one of the six actions "reset to ...° arc" to reset the current arc to a default arc. Some display controls (such as "rim") are also reset.

Select "swap colors" to swap the two surface colors of the current arc. (This is very useful when coloring your sculpture.)

Select "use cols for all" to use the same pair of colors for each arc.

Select "rainbow colors" to color the first arc (or spiral) with a blend of purple and blue, the next with a blend of blue and green, and so on.

Select "alternate colors 1" to reverse the colors of every second arc (or spiral), starting from the first arc.

Select "alternate colors 2" to reverse the colors of every second arc (or spiral), starting from the second arc.

Select action "rotate part by ..." to rotate the current arc or spiral around its center. The from and to angles will be changed accordingly. Click "save" to make the changes permanent.

Select "z-rotate part by ..." to rotate the current arc or spiral around the north-south axis.

Select "start-rotate by ..." to rotate the current arc or spiral around the (green) starting point of your arc or spiral. The angle used for this rotation is displayed right after this drop-down menu.

Select action "end-rotate by ..." to rotate the current arc or spiral around the (red) end point of your arc or spiral. The angle used for this rotation is displayed right after this drop-down menu.

Select one of the six actions "set r1 to ...°" to set the radius of the current arc (or the start radius of the current spiral) to the given fixed level.

Select "resolution -> all" to apply the current resolution to all parts of the sculpture. This will also store the current part as if you had clicked "save".

Select "light = Neutral/Automatic/None" to choose among three different types of lighting.

"center of part at: {.., ..}"

This shows that position of the current arc's (or spiral's) center in polar coordinates (see the next two controls).

Only the integral parts of inclination and azimuth are shown. To see the exact values click the small "+" sign.

"inclination"

Use this slider to define the inclination (angle to the vertical, 0° <= <= 90°) of the center of the currently processed arc.

"azimuth"

Use this slider to define the azimuth (0° <= <= 360°, in the horizontal plane) of the center of the currently processed arc.

Click "arc" if you want a partial circle. Only the first radius slider ("r1") will have an effect.

Click "spiral" if you want to create a (partial) spiral. In this case you will control two radii, one for the (green) starting point of the spiral and one for the (red) end point of the spiral.

" start"

Use this slider to define the radius of the currently processed arc. (Technically the slider defines the inclination of the arc's points relative to the arc's center, as seen from the center of the sphere.)

If a spiral is selected (case 2, see above), then r1 denotes the distance of the (green) starting point to the center of the spiral.

" finish"

Use this slider to define the radius at the (red) end point of the spiral. (Technically the slider defines the inclination between the spiral's points and the spiral's center, as seen from the center of the sphere.)

Use this slider to define the start angle (0–360°) of the currently processed arc or spiral, where = 0° points east and = 90° points north.

Use this slider to define the end angle (0–720°) of the currently processed arc or spiral. This finish value must be larger than the start value.

"resolution"

Use this slider to define the number of polygons in the currently processed arc. The higher this number, the smoother the surface will be.

"join to part"

Use this control to select the arc (or spiral) you want the current arc to join to with a perfectly smooth transition. This target curve must be different from the one currently being edited.

The start and end points of this target will be marked with a green and a red point. Also, the center of the target will be marked by a larger purple point. See details under "gr/gg/rg/rr".

"gr/gg/rg/rr"

Click "gr" to join the green start point of the current arc or spiral to the red end of the target curve. Click "gg" to join the green start point of the current arc to the green start of the target curve. Click "rg" to join the red end point of the current arc to the green start point of the target curve. Click "rr" to join the red end point of the current arc to the red end point of the target curve.

"connected to part: ... via ..."

This displays an existing connection, if there is one.

Note that the information about such a link from arc1 to arc2 is only stored with arc1, not with arc2.

The link information will be deleted from arc1 only if arc2 is deleted, but not when arc2 is moved.

Any arc can only store one such link. That does not limit the number of arcs meeting at one point.

"surface colors from ... to ..."

Each arc or spiral can show a blend of two colors on its surface. The first color is shown near the curve's first vertex, the second color near its last vertex, and a blend of the two colors in between. You can swap the two colors by selecting "swap colors" from the "action" drop-down menu.

Color swatches

Use these controls to select the color of the large sphere ("sphere color") and the "background". Both will be stored together with each sculpture, so different sculptures can have different sphere colors and background colors.

"ribbon"

If the "ribbon" slider is greater than zero, the walls will not go all the way to the center of the sphere. This results in an interesting ribbon effect on the sculpture, and allows you to see through the center of the structure.

"sphere"

Clicking this checkbox makes the large sphere appear/disappear.

"axes"

Clicking this checkbox shows the coordinate axes and the arc's center.

"surface"

Clicking this checkbox hides/shows the surface of the sculpture (which consists of polygons that connect the arcs to the center of the sphere).

"edges"

Clicking this checkbox shows/hides the border lines of the polygons that make up the curved surfaces.

The edges will only be visible if "surface" is activated.

Two "rim" toggles

Clicking the first "rim" checkbox toggles to show/hide the black vertex markers of the current arc.

Clicking the second "rim" checkbox shows a colored outlining of the sculpture (the part that lies on the surface of the surrounding sphere).

It uses the color that is selected for the sphere (see "Color swatches").

"all parts"

Click this checkbox to show all stored pieces at once.

"viewpoint"

Use this control to select different directions from which to view your sculpture.

Hints

• The sculptures "1" to "8" do not use any spirals, only partial circles. Only "9" has spirals.

• As an example of how easy it is to use this Demonstration, we show how to construct sculpture "4" in just a few seconds:

In sculpture "1", select action "reset to 270° arc", then click the buttons "save", "add", "gg", "save", "add", "rr", "save", "add", "gg", "save", "add", "rr", "save", "add", "gg", and "save". That is all.

For the coloring, select the action "rainbow colors", then the action "alternate colors 1".

• In sculpture "2" the direction of every second arc is reversed. So if you want to change the given colors, you do not have to create new colors for every arc.

Simply select one new pair of colors using the swatches and then select the action "use cols for all". The reversed directions of every second arc cause the colors at the seams to match perfectly.

• Working with high resolution can cause the system to be sluggish. Hence, for editing it is recommended that you use low resolution. Once you have finished your basic design, you can increase the resolution for all parts simultaneously by using the action "resolution -> all".

• The states of "sphere color" and all following controls are stored with the sculpture rather than with each arc. The only information not stored at all are lighting and the angle associated with some "actions".

• While the part-independent controls (e.g. "sphere" and "viewpoint") are stored whenever you "save" a part, the part-independent controls are not restored when you use the "restore" control.

• There is no "restore" control for the sculptures, only for arcs. To reverse to the default of a sculpture after editing it, simply select a different sculpture and then re-select the one you were editing.

History

Ilhan Koman is an artist living in Istanbul, Turkey.

Kerim Acar had contacted the author Karl Scherer in April 2011 on behalf of the Koman organization (http://koman.org/).

They were trying to reproduce some works of Ilhan Koman in large scale to be shown at Bogazici University.

Ilhan Ahmet is working to promote science and art all around the world.

Reference

[1] T. Verhoeff and K. Verhoeff, "Lobke, and Other Constructions from Conical Segments," (Jul 15, 2015) archive.bridgesmathart.org/2014/bridges2014-309.html

## Permanent Citation

Karl Scherer

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