Q:

How to create a simple grayscale image with conical gradient?

I need to create an image in illustrator with the following specific requirements. Is this possible? If so, how?
Requirements:
1) The image needs to be in grayscale with a conical gradient such that the values go from 0 at θ = 0, to 255 at θ = 2π.
2) The image needs to be precisely 1920 × 1080 in dimensions.
Thanks in advance!

A:

Conical gradient is tricky to create if you use a pre-CC Illustrator older than CS6 because they have not gradient along a stroke.  If you can place gradient along a stroke, you can simply put it along a circle. There's plenty of examples available, like this.
http://timgthomas.com/2012/05/creating-conical-gradients-in-illustrator-cs6/
A simple workaround with older Illustrators:

draw a rectangle, no stroke, gradient fill
expand the object, release the clipping mask
drag the object to the Brushes collection as a pattern brush, allow it to be stretched to fit
draw a circle, no stroke, no fill
apply the new brush to the circle
expand the appearance
clip the shape to the wanted form and size

You must yourself calculate the needed size and make it at the beginning large enough to avoid the need to stretch it heavily.
You seem to work with pixels. Consider to go to Photoshop, start a new image of size 1920 x 1920. Fill your image with linear gradient. Turn the gradient to conical with Filter > Distort > Polar coordinates. Crop the image to the wanted size with Image > Canvas size. 
Starting with 1920 x 1080 image gives distorted result, the brightness vs angle does not change uniformly. It must be a square.
The result can be imported or copied and pasted to Illustrator, if wanted.

