When you upload the side image of a Cylinder Ttarget to the Target Manager, you must consider the actual shape and circumference of the object. As the Target Manager requires the defined parameters and the uploaded images to have exact dimensions, it is an obligation to properly calculate the shape that you work with.
To make sure that the uploaded images correctly encapsulate the flat body of the real physical object, the flat body-part representing the side surface of a cylinder or cone object is calculated as an rolled out planar surface. The mathematics presented here are particularly relevant when the shape of the object is a cone or a conic frustum. We present the calculations for the conical, cylindrical, and the conical shaped objects.
Dimension Definitions
The following figure shows a generic conic frustum shape, which represents the most general case for conical Cylinder Targets, and its key geometric parameters.
Figure 1: Generic case of conical shape
where
- d and D – are the Bottom and Top diameters of the conical object with d<D and
- sL – is the Side Length of the object.
NOTE: For the general case, the Side Length must not be confused with the cylinder height. However, for the special case of a true cylinder, where the top and bottom diameters are identical and the side surface is aligned vertically, the side length equals the height.
Common Case - the Cylinder
The cylinder is a special case of the generic conical shape since the cylinder has straight edges as shown in the following figure.
Figure 2: Common case cylinder
The top and bottom diameters (d, D) are identical, and the side length simply corresponds to the cylinder height.
Figure 3: Cylinder unrolled layout
Because r' is not defined here, the cylinder’s body layout is a rectangle, where:
- w = πD
- h = sL
NOTE: The flat body must be oriented so that the vertical image axis (dashed line) is parallel to the object vertical axis and it fills the image area.
General Case - Construction of the Flat Conical Body
In the case where top and bottom diameters of a conical body are not identical (i.e. it is not a cylinder), the mantle surface of is not a rectangle when unrolled. To unroll such conical shapes to a rectangular texture, the following considerations need to be taken into account.
Figure 4: A conical shape that is not a cylinder.
The following figure shows two different shapes for the flat side surface. All other instances (cones, for example) are variations on these shapes. The following sections are divided into Case 1 and Case 2, which describe two methods of computing the width and height of the enclosed image.
Figure 5: Two different possible generic cases of a conical object mantle surface (left: D-d < sL right: D-d>=sL)
Case 1 |
Case 2 |
In the following case, the difference between the Top and Bottom diameters is smaller than the Side Length.
Figure 6: Planar construction of body part shape – generic case I |
In this case, the difference between the top and bottom diameters is larger than the Side Length.
Figure 7: Planar construction of body part shape – generic case II |
The flat body can be constructed easily using the given parameters, by applying the following formulas to compute the radii of the two constructing circles:
- Inner circle radius: r' = (d * sL) / (D - d)
- Outer circle radius: R' = r' + sL
For further calculations, use the diameter of the outer circle, which is as follows:
- D' = 2R' = 2 (r' + sL)
The shape can now be constructed by doing the following:
- Draw two concentric circles with radii r' and R'.
- Mark the dashed vertical center line as shown in the figure above.
- Measure the arclength of πD/2 on the outer circle on both sides, from the intersection of the circle with the vertical center line.
- Draw the rays from the two resulting points to the center of the circle.
The shape enclosed between the two circles and the rays is the unrolled surface of the truncated conical shape.
- The circular segment must be oriented in the target image so that it is symmetric on the vertical image axis (the dashed line). The segment borders must touch all four sides of the image.
Width and height of the cylinder’s side image
The width and height of the image for the side surface can be computed from the given parameters and from the computed values for the constructing circles.
In the case where D-d < sL, the width and height of the image are represented by the following equations
- width = D' sin ( πD / D' )
- height = sL + r' ((1 - cos ( πD / D' ))
while in the case where D - d >= sL, the following equations apply:
- width = D'
- height = (D' / 2) ((1 + sin ( (π/2) - (2D - D') / D' ) )
Aspect ratio of the cylinder’s side image
The aspect ratio of the bitmap image can be computed using the following formula:
aspect ratio = bitmapWidth / bitmapHeight = w / h ± 2%
This formula defines that the actual bitmap aspect ratio can deviate by 2% from the required computed ratio. This deviation can be helpful when existing objects are used by cutting, unrolling, and scanning.
Special Case – the Cone
A cone is another special case of a generic conical shape. The cone is characterized by a sharp tip on either the top or the bottom, so that one of the diameters (top or bottom, D, or d) is zero. See the cone example in the following figure.
Figure 8: Special case cone
Figure 9: Cone unrolled layout (left d>sL, right d<sL)
Since one of the diameters of a cone is zero, the formulas can be simplified, as follows:
- r' = 0 and R' = sL – radius of the circle is identical to Side Length, thus
- D' = 2sL
In the case where d < sL, the width of the image is equal to the chord-length of the section:
- w = 2sLsi n ( πd / 2sL ) – width of target image = chord length of section.
while the height of the image is equal to the side length:
- h = sL– height of target image = side length.
In the case where d >= sL, the width of the image is twice the side length:
- w = 2sL– width of target image = twice side length.
while the image height is represented by the following formula:
- h = sL (1 + si n ( π/2 (d - sL) / sL) )
NOTE: The cone segment must be oriented such that the target image is symmetrical on the vertical image axis (the dashed line). The segment borders must touch all four sides of the image.