112 lines
6.6 KiB
Markdown
112 lines
6.6 KiB
Markdown
### Using the Dot Product for Projection {#Heading9}
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Consider Equation 3 again, but this time make one of the vectors, say
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**V**, a unit vector. Now the equation reduces to:
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(eq. 8)
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In other words, the result is the cosine of the angle between the two
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vectors, scaled by the magnitude of the non-unit vector. Now, consider
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that cosine is really just the length of the adjacent leg of a right
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triangle, and think of the non-unit vector as the hypotenuse of a right
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triangle, and remember that all sides of similar triangles scale
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equally. What it all works out to is that the value of the dot product
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of any vector with a unit vector is the length of the first vector
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projected onto the unit vector, as shown in Figure 61.6.
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\
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**Figure 61.6** *How the dot product with a unit vector performs a
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projection.*
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This unlocks all sorts of neat stuff. Want to know the distance from a
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point to a plane? Just dot the vector from the point **P** to the plane
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origin **O**~p~ with the plane unit normal **N**~p~, to project the
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vector onto the normal, then take the absolute value
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distance = |(P - Op) • Np|
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as shown in Figure 61.7.
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Want to clip a line to a plane? Calculate the distance from one endpoint
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to the plane, as just described, and dot the whole line segment with the
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plane normal, to get the full length of the line along the plane normal.
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The ratio of the two dot products is then how far along the line from
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the endpoint the intersection point is; just move along the line segment
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by that distance from the endpoint, and you're at the intersection
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point, as shown in Listing 61.1.
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**LISTING 61.1 L61\_1.C**
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// Given two line endpoints, a point on a plane, and a unit normal
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// for the plane, returns the point of intersection of the line
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// and the plane in intersectpoint.
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#define DOT_PRODUCT(x,y) (x[0]*y[0]+x[1]*y[1]+x[2]*y[2])
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void LineIntersectPlane (float *linestart, float *lineend,
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float *planeorigin, float *planenormal, float *intersectpoint)
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{
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float vec1[3], projectedlinelength, startdistfromplane, scale;
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vec1[0] = linestart[0] - planeorigin[0];
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vec1[1] = linestart[1] - planeorigin[1];
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vec1[2] = linestart[2] - planeorigin[2];
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startdistfromplane = DOT_PRODUCT(vec1, planenormal);
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if (startdistfromplane == 0)
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{
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// point is in plane
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intersectpoint[0] = linestart[0];
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intersectpoint[1] = linestart[1];
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intersectpoint[2] = linestart[1];
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return;
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}
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vec1[0] = linestart[0] - lineend[0];
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vec1[1] = linestart[1] - lineend[1];
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vec1[2] = linestart[2] - lineend[2];
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projectedlinelength = DOT_PRODUCT(vec1, planenormal);
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scale = startdistfromplane / projectedlinelength;
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intersectpoint[0] = linestart[0] - vec1[0] * scale;
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intersectpoint[1] = linestart[1] - vec1[1] * scale;
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intersectpoint[2] = linestart[1] - vec1[2] * scale;
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}
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### Rotation by Projection {#Heading10}
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We can use the dot product's projection capability to look at rotation
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in an interesting way. Typically, rotations are represented by matrices.
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This is certainly a workable representation that encapsulates all
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aspects of transformation in a single object, and is ideal for
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concatenations of rotations and translations. One problem with matrices,
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though, is that many people, myself included, have a hard time looking
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at a matrix of sines and cosines and visualizing what's actually going
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on. So when two 3-D experts, John Carmack and Billy Zelsnack, mentioned
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that they think of rotation differently, in a way that seemed more
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intuitive to me, I thought it was worth passing on.
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\
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**Figure 61.7** *Using the dot product to get the distance from a
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point to a plane.*
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Their approach is this: Think of rotation as projecting coordinates onto
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new axes. That is, given that you have points in, say, worldspace,
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define the new coordinate space (viewspace, for example) you want to
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rotate to by a set of three orthogonal unit vectors defining the new
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axes, and then project each point onto each of the three axes to get the
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coordinates in the new coordinate space, as shown for the 2-D case in
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Figure 61.8. In 3-D, this involves three dot products per point, one to
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project the point onto each axis. Translation can be done separately
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from rotation by simple addition.
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------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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 *Rotation by projection is exactly the same as rotation via matrix multiplication; in fact, the rows of a rotation matrix are the orthogonal unit vectors pointing along the new axes. Rotation by projection buys us no technical advantages, so that's not what's important here; the key is that the concept of rotation by projection, together with a separate translation step, gives us a new way to look at transformation that I, for one, find easier to visualize and experiment with. A new frame of reference for how we think about 3-D frames of reference, if you will.*
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------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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Three things I've learned over the years are that it never hurts to
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learn a new way of looking at things, that it helps to have a clearer,
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more intuitive model in your head of whatever it is you're working on,
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and that new tools, or new ways to use old tools, are Good Things. My
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experience has been that rotation by projection, and dot product tricks
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in general, offer those sorts of benefits for 3-D.
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\
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**Figure 61.8** *Rotation to a new coordinate space by projection onto
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new axes.*
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