How to rotate a part about it's surface?

Having a bit of difficulty understanding what you are trying to do...

I could be completely mistaken, but my understanding is that you want to have the hood of a car rotate up like you are opening it.

For the sake of argument I'll use the example of a chest because I can build a ugly one for this example. To my understanding you want the ability to rotate not from the center of an object, but rather from some other point. In this gif the top of the chest is rotating on it's edge.

To understand how this works we have to understand a little bit about how cframe inverses work. When you multiply a CFrame by it's inverse it becomes the identity CFrame, CFrame.new(). This is super useful b/c the identity CFrame is the equivalent of 1 in CFrame multiplication:

CFrame.Angles(math.pi, math.rad(-14), 0) * CFrame.new(10, -5, 4) * CFrame.new() == CFrame.Angles(math.pi, math.rad(-14), 0) * CFrame.new(10, -5, 4)

Note that the CFrame.new() essentially does nothing to the equality b/c it's like multiplying by 1! So thus say we have some any CFrame, x. The inverse simply means:

x * x:inverse() == x:inverse() * x == CFrame.new()

This is very useful if we treat our CFraming like algebra because it allows us to rearrange things to different sides (since we can't divide cframes).

Say we wanted to solve the following problem. We have two CFrames, A and B. We want to know what we have to multiply against A to get to B. In other words, an offset.

A * offset = B

We know A and B so we are trying to solve for offset. All we have to do is pre-multiply both sides by A:inverse() such that offset is the only thing left on the right hand side.

A:inverse() * A * offset = A:inverse() * B
CFrame.new() * offset = A:inverse() * B
-- Recall: CFrame.new() * offset == offset
offset = A:inverse() * B

If this is all new to you and you are still following then hopefully you will have a mental click moment b/c what we did above to calculate the above is one of the built in CFrame methods. The terms object and world space aren't very helpful to someone who doesn't know what that means, but maybe the above example with offset puts thing into a better perspective!

To continue from here let's step back from CFrames for a second. Say you have a blind friend staying at your house. They want to go out and buy a bag of chips from the local convience store, but of course they're blind so it's not easy for them to go to places they aren't familar with. Unfortunatley you are unavailable to guide them so you have to do your best to describe directions in a way they can navigate it. For someone with eyes and who knows the neighborhood you might just say the intersection where the store is, but that won't work for this friend? What do you do? Well one thing you could do is describe exact directions from where your friend is standing right now. For example, turn 180 degrees, walk 10 steps forward, turn left and walk 20 steps backwards, etc... When we describe directions like this we refer to it as object space because we treat the position that the object starts at as the center of the world.

Going back to CFrames in our case offset is considered object space because it describes the directions that would need to be followed only knowing the starting position/rotation at A to get to B. So in the case of our blind friend we could write a similar equation:

friend_starting_location * directions = store_location

Here's the thing say I change my friends staring location/orientation but still use the exact same directions. Well now I am no longer going to get to the store_location anymore because the directions were an exact set of instructions to get from one specific point to the other:

friend_starting_location * turn_5_degrees_right * directions ~= store_location

To see what I mean let's use a very simply example say my directions to the store are simply take 10 steps forward. Well if I simply adjust the starting location by rotating my freind by say 180 degrees they will no be walking in the complete opposite direction. In fact I can change the starting rotation by any degree and then have them walk forward to have them walk anywhere on a 10 unit radius circle.

friend_starting_location * turn_x_degrees * directions = some_point_on_circle

We can use this exact trick to do the same thing with CFrames.

local lid = game.Workspace.lid;
local edge = lid.CFrame * CFrame.new(lid.Size * Vector3.new(0, -0.5, 0.5))
-- edge * offset = lid.CFrame
local offset = edge:inverse() * lid.CFrame

local r = 0;
while true do
    r = r + math.rad(1);
    lid.CFrame = edge * CFrame.Angles(r, 0, 0) * offset;
    wait();
end;

That's pretty much it! Hope that helped even though it was a very long explanation.

There are a lot of CFrame pages out there on the wiki. I have a few I wrote while back, but idk if they will be helpful or not.

http://wiki.roblox.com/index.php?title=User:EgoMoose/Articles/CFramesAndVectors http://wiki.roblox.com/index.php?title=CFrame_Math

Check out this part of this article b/c it's basically what I was trying to describe above, but it has pictures! http://wiki.roblox.com/index.php?title=CFrame_Math#Rotating_a_door

if P.HingeSurface.Value == "TopSurface" then
               RotationPoint = Part.Position + Vector3.new(0, Part.Size.Y/2, 0)
               Angles = CFrame.fromAxisAngle(Part.CFrame.rightVector, math.pi/4)

If part is rotated so that the "TopSurface" was facing left part.Size.Y/2 would work. This is because if it was rotated 90* and the part isnt Completely square then it wont work because the difference is changing each time

Im not very good at explaining but what im trying to say is that part.Size.Y/2 is not a good way to measure the distance between the surface and its center.

I recommend having an attachment on each face and seeing the distance between the center and that part.

if P.HingeSurface.Value == "TopSurface" then
               RotationPoint = Part.Position + Vector3.new(0, (Part.TopAttachment.Position.Y-Part.Position.Y), 0)
               Angles = CFrame.fromAxisAngle(Part.CFrame.rightVector, math.pi/4)

(IDK if this will actually work my heads not working right. im just saying if the parts rotated then measuring the RotationPoint using the parts size isnt going to work.)

sorry for my explination. hope this helps