Parametric Equations: Deriving (1-t)

In summary, the parametric equations of a line segment from (x0,y0,z0) to (x1,x2,x3) are derived from the formula x = (1-t)x0 + tx1, y = (1-t)y0 + ty1, and z = (1-t)z0 + tz1, where t represents a parameter that ranges from 0 to 1. The (1-t) term is used to interpolate between the initial point (x0,y0,z0) and the final point (x1,x2,x3). This allows for a smooth and continuous representation of the line segment.
  • #1
Marioqwe
68
4
So, the parametric equations of a line segment from (x0,y0,z0) to (x1,x2,x3) are

x = (1-t)x0 + tx1
y = (1-t)y0 + ty1
z = (1-t)z0 + tz1


right?

But how are these equations derived? I don't really see where the (1-t) is coming from.
I would be really thankful if anybody could explain this to me.
 
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  • #2
Hi Marioqwe! :smile:

(try using the X2 icon just above the Reply box :wink:)

It's because it's really

x = x0 + t(x1 - x0)
y = y0 + t(y1 - y0)
z = z0 + t(z1 - z0)

:wink:
 
  • #3
Didn't catch that. Thank you very much.
 

1. What are parametric equations?

Parametric equations are a way of representing mathematical functions using parameters or variables. These equations typically involve the use of t as a parameter, which corresponds to a point on a graph.

2. How do you derive (1-t) in parametric equations?

To derive (1-t) in parametric equations, you can start by thinking of (1-t) as a polynomial. You can then use the power rule to differentiate the polynomial, which would result in the derivative of (1-t) being -1.

3. What is the significance of (1-t) in parametric equations?

The parameter (1-t) in parametric equations represents a point on the graph that is being traced. This allows for more flexibility in representing functions and curves, as the parameter can change and therefore change the point being traced on the graph.

4. How are parametric equations used in real life?

Parametric equations have many real-life applications, particularly in physics and engineering. They are commonly used to describe the motion of objects, such as the trajectory of a projectile or the movement of a pendulum. They can also be used in computer graphics to create smooth curves and animations.

5. What are the advantages of using parametric equations over traditional equations?

Parametric equations offer several advantages over traditional equations. They allow for more flexibility in representing curves and functions, as the parameter can change and therefore change the point being traced on the graph. They also make it easier to describe complex geometric shapes and curves. In addition, parametric equations are often easier to manipulate and solve than traditional equations, making them useful in many real-life applications.

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