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+bjbj22 &DXX"9>>>>>>>R8.4b,Rd(&&&ZZZ\^^^^^^$R>ZZ>>&&46>&>&\\b\lh>>&J(]ƹF^(40@ RR>>>>>4Zn|ZZZRRRRPolygon Clipping Algorithm
Objective
The development of an algorithm, which will transform any polygon to reside within a specified clip region, without any modification of its shape within that region.
Method
The algorithm will scan through a list of connected points, and for each point make sure that it resides within the clip region.
To accomplish this without changing any of the polygons original angels, a point can be divided into two new points, which then are moved inside the current clip region, and thereby keep correct angels of all effected vertices which resides within the clip region.
(NOTE: Explanation of expression: current clip region. As the clip process go through four states, only one clip condition is true in each state. These states are (in order): MinY, MaxY, MinX, and MaxX. For state one, if a points Y-value is greater than or equal to MinY, the point is inside the clip region, whatever its X-value might be. The corresponding condition is true for the three other states.)
Theory
If a point is inside the current clip region, it will be passed to the next state, until all states has been processed. If, however, a point is outside the current clip region, something has to be done.
The current point is outside the current clip region. Lets say that it is point (2), and the near connections then are (1-2-3). The first thing to do is to divide it into two new points, and the connection chain will be (1-2a-2b-3). Now, these new points, (2a) and (2b), must be processed individually, and will either be moved or removed. When processing these points, they are working with a next point, which is point (1) for (2a) and (3) for (2b).
The general rules are simple:
Remove the current point, if next point is outside the current clip region.
Move the current point, if next point is inside the current clip region.
To remove the current point means to disconnect it, and connect its neighbors. If point (2a) were removed, the chain would look like (1-2b-3). If (2b) were removed, the chain would be (1-2a-3), and if both (2a) and (2b), it would be (1-3). You can always remove a divided point that connects to a next point outside the same clip region, as no visible angels will change by doing so, and the extra point wont be needed to represent the clipped polygon inside the clip region.
When moving a point, the new location shall be inside the current clip region and traced on the same line (the line, which intersects the original point and the next point). This is always possible, because in all other cases the point is, instead, removed.
This line and the points new location have some predictable characteristics, which will help simplifying the calculation of the new point:
On horizontal clip region (state one and two), the line will never be horizontal and the points new Y-value will be equal to the clip value (MinY or MaxY).
On vertical clip region (state three and four), the line will never be vertical and the points new X-value will be equal to the clip value (MinX or MaxX).
When the current state is finished, you will to need to choose the next action based on what happened in the state. The following conditions are possible:
Point was inside the clip region. Pass it to next state (unless last state, then restart at state one with the next point in list).
Point was removed. You know nothing about the current point, since it is the next point in list, and therefor you restart at state one (unless it was the last point).
Point(s) was moved. You know that the current point is inside the current and opposite clip region, but the condition relative to the other two is unknown. Either start over with state one, or check these conditions. If in state one, you can skip state two and go directly into state three.
The States
As already mentioned, the procedure involves four similar states, and each involves checking the current point relative to a clip region, and, if necessary, move the point into that clip region. The states can be grouped into horizontal and vertical clip regions.
Here are a list with states and their clip conditions:
Horizontal. Point is inside if its Y-value is greater than or equal to MinY, otherwise outside.
Horizontal. Point is inside if its Y-value is less than or equal to MaxY, otherwise outside.
Vertical. Point is inside if its X-value is greater than or equal to MinX, otherwise outside.
Vertical. Point is inside if its X-value is less than or equal to MaxX, otherwise outside.
A state can be generalized as:
If point is inside clip region, exit and go to next state
Divide current point into two new points with same location.
Handle first and second sub-point: (next point = current 1 and next point = current + 1, respectively)
If next point is outside clip region, remove current point.
If next point is inside clip region, move current point.
If both points was removed, start over at state one
Otherwise point(s) was moved, go to next major state (vertical if on horizontal, and vice versa)
Note that we are all the time working on the same current point. You should think of the current point as the index in the point list. So, if we remove the current point, the next point automatically becomes our current point. It isnt until we reach a fifth state, when we dropped through the four previous states, that we actually start on a new point. As seen in the generalized state above, it is only when the condition in the first step is true in the fourth state, as we reach the fifth state.
It is also a bit tricky in step 3 (handling of sub-points), because if you have moved the first sub-point, current point must be increased before handling the second sub-point. But, when done, current point must be restored to whatever it was before handling the sub-points.
Another important thing to note is that the list of points must be cyclic. If you have a five-vertex polygon, the previous point to point (1) is (5), and the next point to (5) is (1). If you implement this as a list like (5-1-2-3-4-5-1), be sure to update the last (1) when getting there.
Moving Along the Line
When a point is to be moved, its new position is along the line that intersects its original location and the location of its next point. The task is to find the first location on this line that is inside the current clip region. Because this line in general can be of any kind, it is necessary to define different formulas for the two major states (horizontal and vertical).
We know that in a vertical state (three and four), the line can be horizontal but never vertical, and what we need to know is the Y-value (we already know our X-value, which is either MinX or MaxX), and we can therefor base our formulas on the standard line equation:
EQ y = kx + m (Equation 1.1, vertical state line equation)
EQ k = \f( y\s\do3(2) y\s\do3(1) ; x\s\do3(2) x\s\do3(1) ) (Equation 1.2, vertical state line slope equation)
EQ m = y kx (Equation 1.3, vertical state line displacement equation)
In a horizontal state (one and two), the conditions are the opposite. The line can be vertical, but not horizontal, and what we are looking for is the X-value, as the Y-value is either MinY or MaxY. We then have these equations:
EQ x = ky + m (Equation 2.1, horizontal state line equation)
EQ k = \f( x\s \do3(2) x\s\do3(1) ; y\s\do3(2) y\s\do3(1) ) (Equation 2.2, horizontal state line slope equation)
EQ m = x ky (Equation 2.3, horizontal state line displacement equation)
When we are calculating the slope (equation 1.2 and 2.2), x1 and y1 is our current points original location, x2 and y2 is our next point. The same convention is also used in our upcoming trace equation. In the displacement equation, y (equation 1.3) and x (equation 2.3) is derived from the current point, and x (equation 1.3) and y (equation 2.3) is our desired location, which is MinX or MaxX for vertical state, and MinY and MaxY for horizontal state.
What we do is to calculate the slope, k, and the displacement, m, and run them through the line equation along with the desired X- or Y-value. So, we put the slope and displacement equation into the line equations, and we have:
EQ y = y\s\do3(1) + \b \bc\( ( x x\s\do3(1) ) \f( y\s\do3(2) y\s\do3(1) ; x\s \do3(2) x\s\do3(1) ) (Equation 1.4, vertical state trace equation)
EQ x = x\s\do3(1) + \b \bc\( ( y y\s\do3(1) ) \f( x\s \do3(2) x\s\do3(1) ; y\s\do3(2) y\s\do3(1) ) (Equation 2.4, horizontal state trace equation)
These are our final equations. If we look at the first equation (1.4), it will resolve our wanted Y-value for the new location. The variable x is here MinX (state three) or MaxX (state four). As you can see, you can calculate the rightmost term and simply add the result to the original Y-value. In the second equation (2.4) it is the same.
2001 CodeWizard
FILENAME Polygon Clip Algorithm.doc
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