Curves and parametrization

Parametrization refers to the process of expressing relations between variables that depend on each other in terms of independent variables, called parameters. This is useful to describe geometrical objects such as curves and surfaces.

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    Intro

    We all know and love the shape of the DNA-molecule. The DNA-molecule encodes information about who you are and instructions for your body on how to build specific proteins and enzymes vital for your survival.

    The shape of the DNA-molecule was discovered in the 1950's after a long collaboration between chemists, biologists and physicists.

    The helix is an example of a geometric curve that can be parametrized

    The image of a DNA-molecule was captured using a technique called X-ray cristallography and the double helix shape of the DNA-molecule was apparent.

    Concept

    As you're at Disneyland, you ask yourself how you could describe the shape of a roller coaster mathematically

    You could, for example, create a function, modeling the ups and downs of the roller coaster. But that would be quite a... boring roller coaster. You want loops and stuff.

    But in order to describe a loop, you can't use a function.

    Instead, you'll have to use a curve. Then it'll work out. So once you've learned about curves, you're almost a fully-fledged roller coaster pro!

    Math

    The unit circle is described by two variables through

    However, using the following parametrizations

    the unit circle can be described by

    The parametrization for a helix with radius and slope is

    Curve parametrization

    Using parametrization, we can describe almost any curve without more than one variable. This goes for curves not only in but in as well.
    Have a look at this roller coaster, neatly inserted into :

    The position vector is dependent only on the time , so we can write it as:

    This is a parametrization of the curve with as parameter. The position vector of the roller coaster wagon moves along the curve described by the rails as time goes.

    However, curve parametrization doesn't end there: we can use whatever as parameter, depending on what suits our needs. Often, the parameter is one of , or . The parameter doesn't need to have any meaning in the physical world.

    More over, the same curve can be parameterized in endless ways. However, we need to take care to use the right interval for the parameter. The interval depends on which parameter we choose.

    Curve categorization

    To classify curves, we often use the terms closed and simple. It will be useful later to know what this means.

    A closed curve has no ends. It is a loop. A simple curve doesn't intersect itself. If we regard the curves below as lying in , then

    1. is simple, but not closed

    2. is simple and closed

    3. is closed, but not simple

    4. is neither closed nor simple

    Example of curve parametrization

    The semi-circle

    can be parameterized by for example the two following ways:

    To see that these substitutions are correct, substitute the components of the parametrizations into the equation of the semi-circle. The first one becomes:

    The trigonometric identity obtained we know to be true, so the first parametrization holds.

    Likewise, the second parametrization becomes itself as we plug it into the equation to verify it:

    Surface parametrization

    Introduction: the Mercator projection

    The most wide-spread way of visualizing the Earth is called the Mercator projection. It takes every point on the spherical surface of the Earth and maps it onto a plane.

    This mapping has the property that each point on the plane corresponds to exactly one point on the sphere. There is an important distortion happening as we bunch the lines from the north to the south pole together, but it's quite possible to reconstruct the three dimensional globe from the projection map.

    Using surface parametrization, we can describe such transformations. Even though the surface of the Earth is all curvy, it is still a surface. This means that in some coordinate system, this surface will be flat, and we can describe it using only two coordinates.

    Surface parametrization

    We'll cut to the point.

    Let be a continuous function defined on some rectangle in the -plane, having the following coordinates in :

    Then, plugging in all the -pairs in the rectangle into makes up the surface.

    Thus, the surface is actually the range of .

    A side note for the curious: the domain of need not be a rectangle. It just needs to have a well-defined area and consist of an open set and its boundary points. If this confuses you, just forget about it for now.

    It looks messier at first, but the surface parametrization works in the same way as the curve parametrization. The difference is that for the curve, we only needed one variable, because a curve is a one-dimensional object.

    We can regard the surface as two groups of curves, namely those that have a fixed value for and , here called and :

    These lines with different values for the constants are the lines you can see on the surfaces in the picture below.

    Example I

    As a initial example we will parameterize the surface of the sphere. The equation below is a Cartesian equation for a sphere:

    Our goal is to write this equation using two parameters. Not very surprisingly, spherical coordinates are perfect for this task. In spherical coordinates, we have:

    We can plug this all in to the equation, to find what can be:

    In the last two steps, we use trigonometric identities. We now see that and therefore our correct parametrization of the surface is

    Example II

    In the second example we will investigate a paraboloid. Our goal is to parameterize the surface whose equation is:

    We proceed as follows. Observe that the denominators are squares of and respectively. Using this, we set and . Then, we can isolate as a function of and .

    Next we write , and as dependent on the parameters:

    For this problem we could also have used and as parameter. However, introducing and , we could simplify our equation. In problems more advanced this one, it can prove very helpful.

    Surface intersections

    In the wintertime in cold countries, if we're lucky, the lakes freeze. On those frozen lakes, you can see weather-bitten enthusiasts ice skating kilometer after kilometer.

    To test the thickness of the ice as they roam out into the unknown, they use a stick with a sharp end called an ice spike. Shoving it into the ice, it looks as if the surface of the ice 'cuts off' the top of the sharp tip.

    This is what we mean when we talk of surface intersection: the surface of the ice intersects the surface of the cone-shaped ice-spike.

    In multivariable calculus, we're often interested in describing the intersection of two geometric shapes. This intersection is generally made up by a line or curve. In this example here, the intersection is just a circle.

    As with so many other topics in math, it'll become obvious why we perform these kinds of calculations later on. Just wait and see.

    Example

    Find the intersection between the sphere defined by

    and by the plane

    To find the intersection we first isolate in our plane and find it to be

    Next, we plug in to our sphere equation

    Now, we first divide both sides by ten and after that, we use the method called completing the squares:

    Using this expression we can parameterize and . We will use the variable for the parametrization, in order to highlight that we are not using spherical or cylindrical coordinates.

    Next, we can find using the original expression for the plane:

    There are many conventions on writing the parametrization of a surface. Two other common ones are:

    and

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