What is the best way to compute the normal to a parametric surface. The

normal to the parametric surface Q(s,t) is the cross product of the s

tangent vector with the t tangent vector. The s tangent vector is partial

d/ds(Q(s,t)) and the t tangent vector is partial d/dt(Q(s,t)). How should I

evaluate the cross product?

What do you mean: "how should I evaluate"? Are you trying to tell usQuote:> What is the best way to compute the normal to a parametric surface. The

> normal to the parametric surface Q(s,t) is the cross product of the s

> tangent vector with the t tangent vector. The s tangent vector is partial

> d/ds(Q(s,t)) and the t tangent vector is partial d/dt(Q(s,t)). How should I

> evaluate the cross product?

you don't know how the cross product is defined, or what? Look it up

in your textbooks---there's nothing magic about it.

--

Even if all the snow were burnt, ashes would remain.

> What is the best way to compute the normal to a parametric surface. The

> normal to the parametric surface Q(s,t) is the cross product of the s

> tangent vector with the t tangent vector. The s tangent vector is partial

> d/ds(Q(s,t)) and the t tangent vector is partial d/dt(Q(s,t)). How should I

> evaluate the cross product?

probably one doesnt want to use analytical partial derivatives,

because the surface description might be rather complex.

A quadriliteral in the object space (world coordinates) is given

by 4 points Pi(xi,yi,zi), i=1...4 , calculated by the mapping of

the parameter plane si,ti to xi,yi,zi.

Once you decide to use P1 as the reference point, you will find

two edge vectors E21=P2-P1 and E41=P4-P1.

Then calculate the cross product N = E21 x E41 , which is usually

done by a formalistical determinant operation:

i j k

N = E21x E21y E21z

E41x E41y E41z

The normal vector should point outside.

E21y*E41z-E21z*E41y

N = -(E21x*E41z-E21z*E41x)

E21x*E41y-E21y*E41x

Then normalize N by dividing each component by the Euclidian norm of N.

Special attention is necessary for sphere coordinates near to the poles.

Best regards --Gernot Hoffmann

Hi,

To obtain the normal to a parametric surface (at a particular point

s,t) you can sample 3 points on the surface:

Point A = P(s,t)

Point B = P(s+x,t)

Point C = P(s,t+y)

(x and y are small offsets (0.0 < x,y < 1)).

Then use these three points to compute two vectors:

Vector AB

Vector AC

As you noted, the cross product of these two vectors is (a good

approximation of) the normal to the surface at point (s,t).

The smaller the x and y chosen, the more accurate the normal is, but

you may choose x and y proportional to the tesselation rate if you are

using polygons to render the surface.

Neil McLaughlin

On Mon, 16 Dec 2002 14:54:25 +1100, "Jason Heyes"

>What is the best way to compute the normal to a parametric surface. The

>normal to the parametric surface Q(s,t) is the cross product of the s

>tangent vector with the t tangent vector. The s tangent vector is partial

>d/ds(Q(s,t)) and the t tangent vector is partial d/dt(Q(s,t)). How should I

>evaluate the cross product?

1. Normals to parametric surfaces.

What is the best way to compute the normal to a parametric surface. The

normal to the parametric surface Q(s,t) is the cross product of the s

tangent vector with the t tangent vector. The s tangent vector is partial

d/ds(Q(s,t)) and the t tangent vector is partial d/dt(Q(s,t)). How should I

evaluate the cross product?

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