R: Create new Class offspring to "data.frame" with specific constructor

I am trying to create a new class in R for the first time and I have a hard time.

Question

Can you please show me how one could create a class that inherits from the class "data.frame" but can only contain 3 columns named col1, col2 and col3. All columns must be "numeric" and the relation col1 + col2 = col3 must hold true.

The user could do

ClassName(col1 = rep(10,10), col2 = 1:10)

or

ClassName(col1 = rep(10,10), col3 = 11:20)

or

ClassName(col2 = 1:10, col3 = 11:20)

to create an object of this class. All of the above lines would all yield to a similar output to

data.frame(col1 = rep(10,10), col2 = 1:10, col3 = rep(10,10)+(1:10))
   col1 col2 col3
1    10    1   11
2    10    2   12
3    10    3   13
4    10    4   14
5    10    5   15
6    10    6   16
7    10    7   17
8    10    8   18
9    10    9   19
10   10   10   20

What I tried

Here is all I manage to do so far.

setClass("newClass", representation(col1="numeric",col2="numeric",col3="numeric"), contains="numeric")

calc.newClass = function(obj)
{
    if (length(obj@col1)==0)
    {
        return(new("newClass", col1=obj@col3-obj@col2, col2=obj@col2, col3=obj@col3)    )
    }
    if (length(obj@col2)==0)
    {
        return(new("newClass", col1=obj@col1, col2=obj@col3-obj@col1, col3=obj@col3)    )
    }
    if (length(obj@col3)==0)
    {
        return(new("newClass", col1=obj@col1, col2=obj@col2, col3=obj@col1+obj@col2)    )
    }

}

o = calc.newClass(new("newClass", col1=rep(10,10), col3=11:20))

I failed to

• inherit from "data.frame"
• to force the function "calc" to apply automatically at the moment of the construction of an object of the new class.