Develop Reference

A point in 3D space from an angle

I'm currently working on a camera for a game. But I got stuck at the rotation.
When I move my mouse across the x- or y-axis, I want the camera to rotate around my character.
What would be a formula to calculate this vector, if the distance to the character is always the same?
I am doing this in Unity, with C#, if this is of any help.

this function may help: transform.RotateAround(Vector3 axis, float degree)
you can read the Unity Script Reference for more info.
-oh and I think you should tag your next questions with "unity3d"
but you get the best Unity3d-help at UnityAnswers-Forum http://answers.unity3d.com/index.html.

You can utilise spherical coordinates — they seem to fit more than Euler angles for the purpose of camera movement. The Cartesian vector which you need can be obtained by simple formulas as described there.

How to translate a sprite and a vertex geometry by the same distance?

I'm trying to build up a 2.5 engine with XNA. Basically, I want to display a 2D sprites (the main hero and other monsters) in a 3D background. The game will be a platform.
Now, using a translation matrix on a sprite doesn't yield the same result of translate a vertex geometry in world space.
I mean, if I apply
Matrix.CreateTranslation(new Vector3(viewportWidth / 2, viewportHeight / 2, 0));
the sprite will be translate at the middle of screen (starting from the display upper left origin). But, if I apply the same transform to a cube in world space, it will translate very far. This doesn't suprising me, but I wonder of to translate a sprite and a 3D object by the same distance, ignoring all the project/unproject coord stuffs.
Thanks!

There are traditionally three matrices: World, View and Project.
BasicEffect, and most other 3D Effects, simply have those matrices. You use Project to define how points are projected from the 3D world onto the 2D viewport ((-1,-1) in the bottom-left of the viewport to (1,1) in the top-right). You set View to move your camera around in world space. And you use World to move your models around in world space.
SpriteBatch is a bit different. It has an implicit Project matrix that causes your world space to match the viewport's client space ((0,0) in the top-left and (width,height) in the bottom-right). You can pass a transformMatrix matrix to Begin which you can generally think of like the View matrix. And then the parameters you pass to Draw (position, rotation, scale, etc) work like the World matrix would.
If you need to do "weird" things to your World or Project matrices in SpriteBatch, you can just build those transforms into your transformMatrix. It may just involve some maths to "undo" the built-in transformations.
In XNA 4 you can also use an Effect (like BasicEffect) directly in SpriteBatch, which you can provide with arbitrary matrices (details).

Code to extrude 2d geometry to 3d

is there any simple way to extrude a 2d geomtry (vectors ) to a 3d shape
assuming extruding parameter are lenght (double) and angle (degree)
so it should render like a cone ( all z lines going to one point )

(I'd make this a comment, but it's too big)
This isn't just an extruding problem
If it were, your original 2D image would produce either a cylinder with a series of holes in it (not really that useful unless you have a very sophisticated renderer doing either volumetrics or supporting transparency, and the polysort in that case would be very ugly) or 4 cylinders (if I extrude along the inner holes)
most extrusion algorithms don't deal with targeting a single point - that's more than extrusion, it's some form of raycasting
This looks suspiciously like a lighting issue - are you trying to do volumetric lighting, maybe show an effect where the light cone is and deal with the effect of a baffle in front of the light ? Or are you trying to compute the geometry that would define the shadow cast by the object in front of the light ?

Collision Detection for arbitrarily sized, positioned and rotated rectangles in XNA

I'm working with xna in C# and in my game I will have a variety of space ships flying all over the place. They will each have an arbitrary rotation, size and position in space and I need a method to determine when they collide. Ideally the method would take two Rectangles, two doubles and two Vector2s for size, rotation and position respectively and return a boolean that indicates whether they have intersected or not.

Have a look at these links:
Collision Detection Overview
Collision Detection Matrices
Putting Collision Detection Into Practice
They show you a way to do pixel-based collision detection, which is more accurate than rectangle-based for irregularly shaped objects.
Update 2021-01-17 (Martin Senne)
Links of XNA have moved to
XNA
XNA - Collision

You could also consider just using an out of the box solution for this and integrating something like the Farseer Physics Engine:
http://farseerphysics.codeplex.com/

These rectangles you describe are called OBB (Oriented Bounding Boxes)
The way to do collisions between them is using the 'Separating axis theorem'
A really nice page describing it in detail with lots of pictures can be found here

Is there a 3D equivalant to clamp in XNA?

I'm building a 3D game but i only plan on using a 2D perspective thus not taking the z axis into the equasion,
i want to be able to limit the movement of one of my models so it doesn't move out of the unmoving field of view,
when i was designing 2D it was simple just use clamp, but i cant seem to figurebout how to do this in 3d
any help would be much appreciated
Regards

Just use Vector3.Clamp(Vector3 value1, Vector3 min, Vector3 max) and use the constructor on Vector3 that takes a Vector2 and an int (for z value).

If you simply want to do the same thing as a 2D clamp would - Vector3.Clamp set the Y (usually 'UP') component of the two bounding vectors you pass to be 0.
I'm slightly confused on the question however, it seems that maybe what you are after is a form of collision detection with the view frustum This article may help with that if your model can fit into a bounding sphere relativity nicely.
You will need to for test collision vs all the planes which define the view space. If its a perspective camera your using, you'll need to get the Frustum Planes, otherwise if its an orthographic camera they are the planes that make the bounding box of the view space (a cuboid which is orientated the same way as the camera).

So you want in pseudocode:
if (object.position+movementVec is in view of camera)
object.position+=movementVec;