To explain the title, consider these two captures. The first one is a sphere that is centered at (0, 0, 0) which is the target of the Matrix.CreateLookAt with target being (0, 0, 0). In the second, the same sphere was moved to the side (Vector3.Right * some factor). The sphere is now somewhat skewed and oval-looking. This is due to the properties of perspective field of view.

I wish to render the sphere to the side as in the second capture, but with the same perspective as in first. One thing I tried is to change the field of view to a smaller, near orthographic, value. This solves the problem but creates other problems, where other objects are now subject to what is effectively a near-orthographic view. Another option is to capture the sphere into a render target and draw it separately. The problem with this is that the sphere will no longer be in 3D space. Furthermore, unnecessary GPU resources will be committed just for this, impacting performance.

Any other ideas on how I can achieve this?

It isn't really possible, mathematically, to have both the properties of perspective you want (things farther away are smaller) without the properties of perspective you don't (objects close to the focal point and off to the side are noticeably distorted). At least not when you've got actual 3D space involved.

You could:

1. Render each object individually with the camera looking directly at it and composite these elements into a flat, orthographic scene, and hack the size change due to distance by scaling each relative to its 'distance' from the camera. Kind of like billboards. You can still use the GPU to depth sort them, but you'd be manipulating the depth manually.
2. Skew the perspective matrix such that the screenspace (x,y) of the focal point is always aligned to the screenspace (x,y) position of the sphere. You will have to manually manipulate the matrices, and they have to be homogenous. It will also cause disorienting shifts in objects behind and in front of the sphere as the sphere moves.

I'm curious, why do you want to do this? I think your eye does the same thing to some extent so it's actually less realistic.

26 minutes ago, L. Spiro said:

Your eyes do not distort images that are off-to-the-side this way.

Now that I think about it more, I think you are right because the receptors on the back of your eye are roughly on the inside of a sphere. However I would think there should be some way to compensate for it by mapping pixels in the same way.

Be interesting to try it though. Maybe you could simply map the (x,y) coordinates of the pixels. by their angle from the center line in the X and Y directions.  I'm not sure you need ray-tracing for this. I may try when I get some extra time since I'm working on DirectX shaders at the moment.

gpu intensive but... increase distance param in engine somehow so you're looking at objects from further away and then only show the mid 80% of screen but w/zoom to take up your whole screen. the off-angle distortion would be  greatly reduced and perhaps feel more organic? 
Lol... i work w/2d stuff primarily so tend to have a lot of cpu cycles to spare.... probably not the case in your game.

I think  maybe you can just take x/z and y/z, round them to the appropriate level, and then plug those into an arctan table.

Thanks for the replies. @L. Spiro, to address your point of "...this is not an issue.  There is nothing here to be fixed."... correct, there's nothing wrong here. I just need a specific object to be render in this manner due to the nature of the game I am working.

With that said, I think the only two solutions that I can consider is point 1. that @Zorinthrox suggested, or to capture it into a render target.