How do i Warp a cylinder into a flat rectangle

While, on the one hand, I am very reluctant to suggest a solution I haven’t personally tried, on the other, I am tempted to tell you what I would try if I had the opportunity. It is this:

Photography:

Both camera and object on a vertical axis with the lens vertical and horizontal axis bisecting the object. Lighting tented to avoid blue hot spots appearing in highlights. Longer than normal focal length lens. Small -- not smallest -- aperture. 

The image above violates these shooting criteria but I used them to display the method because of the object's shape. The result of this technique should be evaluated in that context. For each exposure, rotate the object and assume you will be using only the middle third of the object in final image assembly.

Photoshop:

Perspective-Transform to expand the top half of each image and contract the lower half of the image to create a rectangle shape (head-on cylinder). This will avoid dead space in the areas adjacent to the neck of the vase. Expand horizontally and vertically to decompress the object and compensate for compression in the Transform command.  Final crop from top and stetch base to aid composition. (If the object is perfectly symmetrical, you may be able to stack the images as layers and apply the Transform to all rather than individually.) 

Hi

You didn't say how you photographed the image.

• Did you use a tripod and have lots of overlap?
• Which settings (plural) did you try for the pano?
• Did you move the camera or the object?

https://helpx.adobe.com/photoshop/using/create-panoramic-images-photomerge.html

~ Jane

A suggestion I read recently was Put your vase on a turntable, and photograph it with the panorama setting on a smartphone.