Sudipta Mondal

Computer Graphics 9:Clipping In 3D

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3-D Clipping

Just like the case in two dimensions,clipping removes objects that will not bevisible from the sceneThe point of this is to remove computationaleffort3-D clipping is achieved in two basic steps

– Discard objects that can’t be viewed• i.e. objects that are behind the camera, outside

the field of view, or too far away– Clip objects that intersect with any clipping

plane

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Discard Objects

Discarding objects that cannot possibly beseen involves comparing an objectsbounding box/sphere against thedimensions of the view volume

– Can be done before or after projection

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Clipping Objects

Objects that are partially within the viewingvolume need to be clipped – just like the 2Dcase

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The Clipping Volume

After the perspective transformation iscomplete the frustum shaped viewingvolume has been converted to aparallelopiped - remember we preserved allz coordinate depth information

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Normalisation

The transformed volume is then normalisedaround position (0, 0, 0) and the z axis isreversed

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When Do We Clip?

We perform clipping after the projectiontransformation and normalisation arecompleteSo, we have the following:

We apply all clipping to these homogeneouscoordinates

úúúú

û

ù

êêêê

ë

é

×=

úúúú

û

ù

êêêê

ë

é

1zyx

M

hzyx

h

h

h

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Dividing Up The World

Similar to the case in two dimensions, wedivide the world into regionsThis time we use a 6-bit region code to giveus 27 different region codesThe bits in these regions codes are asfollows:

bit 6Far

bit 5Near

bit 4Top

bit 3Bottom

bit 2Right

bit 1Left

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Dividing Up The World (cont..)

Because we have a normalised clippingvolume we can test for these regions asfollows:

Rearranging these we get:

11 ££-hxh 11 ££-

hyh 11 ££-

hzh

hxh h ££- hyh h ££- hzh h ££- 0>hif

hxh h -££ hyh h -££ hzh h -££ 0<hif

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Region CodesIm

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Point Clipping

Point clipping is trivial so we won’t spendany time on it

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Line Clipping

To clip lines we first label all end points withthe appropriate region codesWe can trivially accept all lines with bothend-points in the [000000] regionWe can trivially reject all lines whose endpoints share a common bit in any position

– This is just like the 2 dimensional case asthese lines can never cross the viewingvolume

– In the example that follows the line fromP3[010101] to P4[100110] can be rejected

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Line Clipping ExampleIm

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The Equation Of The Line For 3DClipping

For clipping equations for three dimensionalline segments are given in their parametricformFor a line segment with end points P1(x1h,y1h, z1h, h1) and P2(x2h, y2h, z2h, h2) theparametric equation describing any point onthe line is:

uPPPP )( 121 -+= 10 ££ u

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The Equation Of The Line For 3DClipping (cont…)

From this parametric equation of a line wecan generate the equations for thehomogeneous coordinates:

uhhhhuzzzzuyyyyuxxxx

hhhh

hhhh

hhhh

)12(1)12(1)12(1)12(1

-+=-+=-+=-+=

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3D Line Clipping Example

Consider the line P1[000010] to P2[001001]Because the lines have different values in bit2 we know the line crosses the right boundary

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3D Line Clipping Example (cont…)

Since the right boundary is at x = 1 we nowknow the following holds:

which we can solve for u as follows:

using this value for u we can then solve for ypand zp similarly

1)12(1

)12(1=

-+-+

==uhhh

uxxxhxx hhhh

p

)22()11(11

hxhxhxu

hh

h

----

=

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3D Line Clipping Example (cont…)

When then simply continue as per the twodimensional line clipping algorithm

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3D Polygon Clipping

However the most common case in 3Dclipping is that we are clipping graphicsobjects made up of polygons

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3D Polygon Clipping (cont…)

In this case we first try to eliminate the entireobject using its bounding volumeNext we perform clipping on the individualpolygons using the Sutherland-Hodgmanalgorithm we studied previously

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Cheating with Clipping Planes

For far clipping plane introduce something toobscure far away objects – fogMake objects very near the cameratransparent