Can somebody help me with glTexCoord2d?

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Let's say I want to create a huge polygon that is basically a 2D terrain. (That means that the points on the top of the object will vary in height, but on the bottom they will all be zero.)

Can I use this polygon as a cookie-cutter on a texture? (eg. my texture image will fill my polygon and will not repeat, etc)

I'm thinking I have to use a bunch of glTexCoord2d() calls in-between glVertex2f() calls but I'm not sure.

If this method won't work, will I have to create a TON of quads 1-pixel in width?

Thanks for your help!