What formula or algorithm can I use to draw a 3D sphere without using an OpenGL type library?

I know that there are 4 techniques to draw 3D objects:

(1) Wireframe Modeling and rendering, (2) Additive Modeling, (3) Subtractive Modeling, (4) Splines and curves.

Then, those models go through hidden surface removal algorithm.

Am I correct?

Be that way, What formula or algorithm can I use to draw a 3D Sphere?

I am using a low-level library named WinBGIm from colorado university.

there are 4 techniques to draw 3D objects:

(1) Wireframe Modeling and rendering, (2) Additive Modeling, (3) Subtractive Modeling, (4) Splines and curves.

These are modelling techniques and not rendering techniques. They allow you to mathematically define your mesh's geometry. How you render this data on to a 2D canvas is another story.

There are two fundamental approaches to rendering 3D models on a 2D canvas.

Ray Tracing

The basic idea of ray tracing is to pass a ray from the camera's origin, through the point on the canvas whose colour needs to be determined. Determine which models get hit by it and pick the closest one, determine how it's lit to compute the colour there. This is done by further tracing rays from the hit point to all the light sources in the scene. If you notice, this approach eliminates the need to use hidden surface determination algorithms like the back face culling, z-buffer, etc. since the basic idea is rooted on a hidden surface algorithm (ray tracing).

There are packages, libraries, etc. that help you do this. However, it's common that ray tracers are written from scratch as a college-level project. However, this approach takes more time to render (not to code), but the results are generally more pleasing than the below one. This approach is more popular when you want to render non-interactive visuals like movies.

Rasterization

This approach takes primitives (triangles and quads) that define the models in the scene and sample them at regular intervals (screen pixels they cover) and write it on to a colour buffer. Here hidden surface is usually eliminated using the Z-buffer; a buffer that stores the z-order of the fragment and the closer one wins, when writing to the colour buffer.

Rasterization is the more popular approach with cheap hardware support for it available on most modern computers due to years of research and money that has gone in to it. Libraries like OpenGL and Direct3D are readily available to facilitate development. Although the results are less pleasing than ray tracing, it's faster to render and thus is widely used in interactive, real-time rendering like games.

If you want to not use those libraries, then you have to do what is commonly known as software rendering i.e. you will end up doing what these libraries do.

What formula or algorithm can I use to draw a 3D Sphere?

Depends on which one of the above you choose. If you simply rasterize a 3D sphere in 2D with orthographic projection, all you have to do is draw a circle on the canvas.