Introduction
A ruled surface is a surface between two curves connected by a straight line. The connected straight lines also have to lie on the surface. These lines would be made for all the points on the curves. To think about this, we can look at a tire laid flat on the ground. We know that there are two curves, a circle on top and a circle on the bottom. Using the definition, we know that the tire would be considered a ruled surface because we are able to make straight lines around both the curves connecting the two circles, where the lines always lie on the surface. In general, this example is just a cylinder but we can also have a ruled surface to form hyperboloids, cones and many more shapes.
Ruled Surface Example
Let's look at another simple example, a ladder. A ladder is made up of two straight lines, which are the curves. The steps on the ladders can be seen as connecting the two curves, which we know are straight that connect the outer two curves. This would also be considered a ruled surface. Now let's look at the same ladder example but let's say we twist one of the outer lines around the other line. You might think that the lines would not be straight but, in the image, below we can see it looks like a spiral staircase, also meaning that the lines are straight and lie on the surface. We can twist our curves as long as the lines we make are still straight. These are all examples that can seem boring, now we can look at something more interesting.
Our object is in the xyz-plane so I will be parametrizing each of the curves separately to give us a better idea of what is happening with the shape we get. For our first curve we get \begin{align*} x(s)&= \cos(s) + \cos^3(s) \\ y(s)&= \sin(s) + \sin^3(s) \\ z(s)&= 10 \end{align*}
These equation from above form the curve that can be seen on the right. For each value of s, we would plug it into the equations above. This would give us a point on the xyz-plane. This is our top curve.
Our bottom curve is given by the set of equations below. These equations make the curve on the right. We can differentiate which curve is on the top by the z(s) equations which are both constants. \begin{align*} x(s)&= \cos(s) + \cos^4(s) \\ y(s)&= \sin(s) + \sin^3(s) \\ z(s)&= 0 \end{align*}
These are our two curves now let's view them as a ruled surface. As we recall we should be able to make straight lines from the bottom curve to the top. In the image below on the left we can see that this would be considered a ruled surface as we can see the straight lines connecting the two curves. However, on the right we can see the same curves, but the top curve is shifted by 100 degrees. This may seem confusing, but if we look closely the lines connecting the curves are still straight, but now they are slanted. Both surfaces are considered as ruled surfaces. We can twist our curve further, but we would then get lines that intersect and it wouldn't look too appealing.
One additional thing that should be noted is that there is no limit on how many lines that can be used to connect the curves. In the next image you can see that there are much more lines used than the previous images to connect the curves.
Why This Example
The parametrization for both curves are very similar. Which makes it very interesting because our two curves look very different from each other. I also like this example because we are able to see the lines connecting the two curves are slanted but are still straight. When I was choosing my equations the final outcome of these reminded me of a vase, probably one you can find on Valentine's Day. Even after picking interesting curves it is always helpful being able to visualize the final product. The dimensions of this object should be 29.53mm × 38.58mm × 45.56mm when printed.
A ruled surface is a surface between two curves connected by a straight line. The connected straight lines also have to lie on the surface. These lines would be made for all the points on the curves. To think about this, we can look at a tire laid flat on the ground. We know that there are two curves, a circle on top and a circle on the bottom. Using the definition, we know that the tire would be considered a ruled surface because we are able to make straight lines around both the curves connecting the two circles, where the lines always lie on the surface. In general, this example is just a cylinder but we can also have a ruled surface to form hyperboloids, cones and many more shapes.
Ruled Surface Example
Let's look at another simple example, a ladder. A ladder is made up of two straight lines, which are the curves. The steps on the ladders can be seen as connecting the two curves, which we know are straight that connect the outer two curves. This would also be considered a ruled surface. Now let's look at the same ladder example but let's say we twist one of the outer lines around the other line. You might think that the lines would not be straight but, in the image, below we can see it looks like a spiral staircase, also meaning that the lines are straight and lie on the surface. We can twist our curves as long as the lines we make are still straight. These are all examples that can seem boring, now we can look at something more interesting.
Our object is in the xyz-plane so I will be parametrizing each of the curves separately to give us a better idea of what is happening with the shape we get. For our first curve we get \begin{align*} x(s)&= \cos(s) + \cos^3(s) \\ y(s)&= \sin(s) + \sin^3(s) \\ z(s)&= 10 \end{align*}
These equation from above form the curve that can be seen on the right. For each value of s, we would plug it into the equations above. This would give us a point on the xyz-plane. This is our top curve.
Our bottom curve is given by the set of equations below. These equations make the curve on the right. We can differentiate which curve is on the top by the z(s) equations which are both constants. \begin{align*} x(s)&= \cos(s) + \cos^4(s) \\ y(s)&= \sin(s) + \sin^3(s) \\ z(s)&= 0 \end{align*}
These are our two curves now let's view them as a ruled surface. As we recall we should be able to make straight lines from the bottom curve to the top. In the image below on the left we can see that this would be considered a ruled surface as we can see the straight lines connecting the two curves. However, on the right we can see the same curves, but the top curve is shifted by 100 degrees. This may seem confusing, but if we look closely the lines connecting the curves are still straight, but now they are slanted. Both surfaces are considered as ruled surfaces. We can twist our curve further, but we would then get lines that intersect and it wouldn't look too appealing.
Why This Example
The parametrization for both curves are very similar. Which makes it very interesting because our two curves look very different from each other. I also like this example because we are able to see the lines connecting the two curves are slanted but are still straight. When I was choosing my equations the final outcome of these reminded me of a vase, probably one you can find on Valentine's Day. Even after picking interesting curves it is always helpful being able to visualize the final product. The dimensions of this object should be 29.53mm × 38.58mm × 45.56mm when printed.
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