What is the ruling on photographing animate beings and hanging the pictures, and what is the ruling on my work photographing show jumping championships in equestrianism, given the presence of a قطع للجسم (part of a body—the rider or the horse) in every picture?

Scholars have differed concerning the ruling on photographing animate beings, some permitting it and others prohibiting it. If it is said to be permissible, then there is no problem with your work. But if it is said to be forbidden, then it is not permissible. The presence of a cutter for the body does not change the ruling. Rather, what removes the picture from prohibition is for it to be incomplete with a deficiency that makes the depicted subject unable to live.