# Orthogonal projection of 2 points onto a plane

• Bozebo
In summary, the author is asking for help understanding a maths problem, but has not progressed far enough to solve it. They are asking for help and are scared of getting lost.
Bozebo

Hi, I have ended up on this site a few times after searching various maths issues; it seems to have a good community so I am asking you good people for a little help understanding this.

Tomorrow I have a semi-important maths exam, if I fail it I can resit in 2010 but I would of course rather not have to do that. I am studying game design, so I shall be needing maths a lot more soon.

I missed the part that involved the mathematics of projection, so I need to figure this out fast.
Reading the materials I understood fairly well until it bluntly proposed a problem without before giving a solution to a similar one, so I need a bit of help.

The problem is as follows:
Calculate the orthogonal projection of the points A(3,5,1) and B(-4,2,6) onto the plane -2x+4y-z=3 and determine the equation of the line joining the image points on the plane.

I understand visually what is going on here, but I don't know where to start. The notes talk about the Par i/j/k matrices (none seem to apply here) and the homogeneous coordinate transformation matrix... now I could plug numbers in and see what I come up with, its not like I am going to run out of paper. But I am scared to get lost in this.

Who can help walk me through this? It may take a few posts ^_^edit:

vector n = (-2,4,-1)

n being the normal
brackets should be vertical... but I don't know how to format it that way

edit:
Ok, I have got this far:

-2(3-2t)+4(5+4t)-(1-t)=3

correct so far? I filled in the values for p=r+tn

Last edited:
Bozebo said:

Hi, I have ended up on this site a few times after searching various maths issues; it seems to have a good community so I am asking you good people for a little help understanding this.

Tomorrow I have a semi-important maths exam, if I fail it I can resit in 2010 but I would of course rather not have to do that. I am studying game design, so I shall be needing maths a lot more soon.

I missed the part that involved the mathematics of projection, so I need to figure this out fast.
Reading the materials I understood fairly well until it bluntly proposed a problem without before giving a solution to a similar one, so I need a bit of help.

The problem is as follows:
Calculate the orthogonal projection of the points A(3,5,1) and B(-4,2,6) onto the plane -2x+4y-z=3 and determine the equation of the line joining the image points on the plane.
A normal vector to -2x+ 4y- z= 3 is <-2, 4, -1>. A line through A(3,5,1) in that direction (and so normal to the plane) is given by x= 3- 2t, y= 5+ 4t, z= 1- t. Put those into the equation of the plane to find the point where that line crosses the plane. Do the same with B(-4,2,6): x= -4- 2t, y= 2+ 4t, z= 6- 6.

I understand visually what is going on here, but I don't know where to start. The notes talk about the Par i/j/k matrices (none seem to apply here) and the homogeneous coordinate transformation matrix... now I could plug numbers in and see what I come up with, its not like I am going to run out of paper. But I am scared to get lost in this.

Who can help walk me through this? It may take a few posts ^_^

edit:

vector n = (-2,4,-1)

n being the normal
brackets should be vertical... but I don't know how to format it that way

edit:
Ok, I have got this far:

-2(3-2t)+4(5+4t)-(1-t)=3

correct so far? I filled in the values for p=r+tn

OK, I've found the co-ordinates of the projection of point A. But they seem odd to me.
I have:
x = 1/3
y = 10 and 1/3
z = -1/3

I solved t to 4/3, does that seem right?

That's not what I get- what I get is even more peculiar. Show how you solved the equation, please.

OK here we go.

Calculate the orthogonal projection of points A(3,5,-1) and B(-4,2,6) onto the plane -2x+4y-z=3

r = a + tb
<X,Y,Z> = <3,5,-1> + t<-2,4,-1>

X = 3 - 2t
Y = 5 + 4t
Z = -1 -t

aX +bY +cZ = d
-2(3 - 2t) + 4(5 + 4t) - ((-1) - t) = 3

t ends up as -18/21 this time

I calculated X of the projection of point A to be 9/7 - is it good this time?

Now, can I use any of those results to calculate the projected point B? because there is a problem later on confusing me that asks "A camera is aligned such that it projects the point (1,1,2) to the origin by an orthographic projection. Determine the image of the point (-1,4,-3) under the same projection". Any hints for where to start this one?

Bozebo said:
OK here we go.

Calculate the orthogonal projection of points A(3,5,-1) and B(-4,2,6) onto the plane -2x+4y-z=3

r = a + tb
<X,Y,Z> = <3,5,-1> + t<-2,4,-1>

X = 3 - 2t
Y = 5 + 4t
Z = -1 -t

aX +bY +cZ = d
-2(3 - 2t) + 4(5 + 4t) - ((-1) - t) = 3
So -6+ 4t+ 20+ 16t+ 1+ t= 3
(4+16+1)t= 3+ 6- 20- 1
21t= -12
t= -12/21= -4/7

t ends up as -18/21 this time

I calculated X of the projection of point A to be 9/7 - is it good this time?

Now, can I use any of those results to calculate the projected point B? because there is a problem later on confusing me that asks "A camera is aligned such that it projects the point (1,1,2) to the origin by an orthographic projection. Determine the image of the point (-1,4,-3) under the same projection". Any hints for where to start this one?

## 1. What is an orthogonal projection of 2 points onto a plane?

An orthogonal projection of 2 points onto a plane is a mathematical operation that involves finding the shortest distance between two points and a plane. It is used to determine the position of a point in relation to a plane and is a common technique used in engineering, computer graphics, and physics.

## 2. How is an orthogonal projection of 2 points onto a plane calculated?

The calculation for an orthogonal projection involves finding the dot product between the vector representing the line connecting the two points and the normal vector of the plane. This dot product is then divided by the magnitude of the normal vector to find the distance between the plane and the line. The result is then used to find the coordinates of the projected point on the plane.

## 3. What is the purpose of performing an orthogonal projection of 2 points onto a plane?

The purpose of an orthogonal projection is to determine the shortest distance between two points and a plane. This information is useful in various fields, such as architecture, where it is important to know the distance between a building and the ground or in computer graphics to create realistic 3D objects.

## 4. Can an orthogonal projection of 2 points onto a plane be negative?

Yes, an orthogonal projection can be negative. This occurs when the line connecting the two points is on the opposite side of the plane's normal vector. In this case, the distance between the plane and the line will be negative, indicating that the two points are on opposite sides of the plane.

## 5. Is an orthogonal projection of 2 points onto a plane affected by the orientation of the plane?

Yes, the orientation of the plane can affect the result of an orthogonal projection. The normal vector of the plane is used in the calculation, so if the plane is rotated or flipped, the normal vector will change, resulting in a different projection. It is important to carefully consider the orientation of the plane when performing an orthogonal projection.

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