Drawing Arcs Efficiently On ZX Spectrum Using Bresenham's Algorithm

Hey guys! So, you're diving into the awesome world of retro graphics on the ZX Spectrum and need a way to draw arcs, huh? You've come to the right place! Drawing arcs can be a bit tricky, especially when you're working with the Z80 assembler's limitations. But don't worry, we'll explore some super-efficient algorithms to get those perfect curves on your screen. Let's get started!

Why Arcs are a Must-Have in Your Graphics Toolkit

First off, let's talk about why arcs are so important. Think about it – circles, curves, and rounded shapes make graphics look way more polished and professional. Imagine trying to draw a car, a character's face, or even just a simple button without being able to draw an arc. It would be a pixelated nightmare! Having a fast, reliable arc-drawing function in your arsenal opens up a whole new world of possibilities for your ZX Spectrum games and demos.

When we talk about drawing arcs efficiently, especially on a machine like the ZX Spectrum, the algorithm you choose is super crucial. We need something that minimizes calculations because every clock cycle counts! That's why we'll be focusing on algorithms that use integer arithmetic and avoid those slow floating-point operations. Believe me, your Z80 will thank you!

Now, you might be thinking, “Why not just use a lookup table?” That's a valid idea, and it can work for certain situations. But lookup tables take up precious memory, and on the ZX Spectrum, memory is a resource we can't afford to waste. Plus, if you need arcs of different radii, you'd need multiple lookup tables, which compounds the problem. So, an algorithmic approach is often the best way to go for flexibility and memory efficiency.

And let’s be real, writing these kinds of routines is just plain fun! It’s a chance to get down and dirty with the hardware, optimize your code, and really understand how graphics work at a low level. There’s a certain satisfaction that comes from seeing those perfectly formed arcs appear on the screen, knowing you crafted every pixel with your own code.

So, buckle up, folks! We're about to dive into the heart of arc-drawing algorithms. We'll start with the classic Bresenham's circle algorithm and then explore some alternative approaches. By the end of this guide, you'll be drawing arcs like a pro on your ZX Spectrum!

Bresenham's Circle Algorithm: The Classic Approach

Let's get into the heart of the matter: Bresenham's circle algorithm. This algorithm is a total classic for a reason. It's fast, it's efficient, and it only uses integer arithmetic – perfect for our Z80! The core idea behind Bresenham's algorithm is to make decisions about which pixel to draw next based on an error term. We're essentially approximating a circle by stepping through the pixels in a clever way, minimizing the distance between the plotted pixels and the true circle.

Okay, so how does it actually work? Imagine we're drawing a circle centered at (0, 0). We only need to calculate the pixels for one octant (an eighth) of the circle, typically from 0 to 45 degrees. Why? Because we can use symmetry to reflect those pixels and get the other seven octants! This trick dramatically reduces the amount of computation we need to do. Pretty neat, huh?

We start at the point (0, r), where r is the radius of the circle. Our goal is to move step-by-step towards the 45-degree line, deciding whether to move one pixel to the right (increment x) or one pixel diagonally (increment both x and y) at each step. The decision is based on an error term, which represents how far away we are from the true circle. If the error term is positive, it means we've gone too far outside the circle, so we move only horizontally. If it's negative, we're still inside the circle, so we move diagonally to get closer to the edge.

The error term is updated at each step based on the decision we made. This is where the integer arithmetic magic happens. The updates involve only additions and subtractions, which are super-fast operations on the Z80. We completely avoid multiplication and division, which would slow things down considerably.

Now, let's talk about the ZX Spectrum's quirks. The screen is organized in a peculiar way, with the attribute clash issue lurking around every corner. This means that writing to the screen can be slower in certain areas where different colors collide. To optimize our arc-drawing routine, we need to be mindful of these attribute clashes. One strategy is to draw the arc in a way that minimizes the number of attribute changes. For example, drawing in a solid color can help avoid these clashes. We might even consider buffering the pixel data in memory and then writing it to the screen in one go, which can be more efficient in some cases.

Bresenham's algorithm provides a solid foundation, but there's always room for optimization. We can unroll loops, use pre-calculated values, and employ clever bit-shifting tricks to squeeze every last drop of performance out of the Z80. It’s a fun challenge, and the results are well worth the effort!

Beyond Bresenham: Exploring Alternative Arc-Drawing Algorithms

While Bresenham's algorithm is a total workhorse, it's always good to have options, right? There are other arc-drawing algorithms out there that might be a better fit for your specific needs or coding style. Let's explore some alternatives and see what they bring to the table.

One popular alternative is the Midpoint Circle Algorithm, which is very similar to Bresenham's but uses a slightly different way of calculating the error term. The core concept remains the same: we're making pixel-by-pixel decisions based on an error term to approximate the circle. The Midpoint algorithm also boasts the same advantages as Bresenham's: it uses only integer arithmetic and leverages symmetry to reduce computation.

The main difference lies in how the error term is calculated and updated. In the Midpoint algorithm, we evaluate the midpoint between two potential pixels (one horizontal and one diagonal) to determine which pixel is closer to the true circle. This slight change in approach can sometimes lead to minor performance differences or code readability preferences. Some developers find the Midpoint algorithm easier to understand or implement, while others prefer Bresenham's. It's really a matter of personal choice!

Another interesting approach is the parametric circle equation. Instead of stepping through pixels in a grid-like fashion, we can calculate the x and y coordinates of points on the circle using trigonometric functions (sine and cosine). The equation looks like this: x = r * cos(theta), y = r * sin(theta), where r is the radius and theta is the angle. By varying theta, we can generate points along the circle.

Now, you might be thinking, “Wait a minute, trigonometric functions? That sounds like floating-point math!” And you're right, directly using sine and cosine would be a performance killer on the Z80. But there are clever ways to approximate these functions using integer arithmetic. One common technique is to use lookup tables containing pre-calculated sine and cosine values for a limited set of angles. We can then interpolate between these values to get a reasonable approximation for any angle.

The parametric approach can be particularly useful if you need to draw partial arcs or arcs with specific start and end angles. It gives you more direct control over the arc's shape and position. However, it generally involves more calculations than Bresenham's or the Midpoint algorithm, so it might not be the best choice if performance is your top priority.

There are also more specialized algorithms out there, such as those based on polynomial approximations or other mathematical techniques. These algorithms might offer certain advantages in specific situations, but they often come with added complexity. For most general-purpose arc-drawing tasks on the ZX Spectrum, Bresenham's algorithm or the Midpoint algorithm will provide the best balance of performance and simplicity.

Optimizing Your Arc-Drawing Routine for the ZX Spectrum

Alright, so you've got your arc-drawing algorithm down, but the real magic happens when you start optimizing it for the ZX Spectrum. This little machine has its quirks and limitations, but with some clever tricks, we can squeeze out every last bit of performance. Let's dive into some key optimization strategies.

First and foremost, think about integer arithmetic. We've already talked about how Bresenham's and the Midpoint algorithm use integer math to avoid slow floating-point operations. But even within integer arithmetic, there are faster and slower operations. Additions, subtractions, and bit shifts are your friends. Multiplications and divisions? Not so much. Try to rewrite your equations to use the faster operations whenever possible.

Another critical aspect is loop unrolling. Loops are a fundamental part of any arc-drawing algorithm, but they also introduce overhead. Each time the loop iterates, the Z80 has to execute instructions for incrementing the loop counter, checking the loop condition, and jumping back to the beginning of the loop. By unrolling the loop, we can reduce this overhead. Unrolling means manually repeating the loop's body multiple times within the code, effectively reducing the number of loop iterations. This can make your code longer, but it can also make it significantly faster.

Lookup tables can also be your allies in the quest for speed. We mentioned earlier that using trigonometric functions directly is slow, but we can pre-calculate sine and cosine values and store them in a table. Then, instead of calculating these functions on the fly, we can simply look up the values in the table. This can be a huge win for performance, especially if you're using the parametric circle equation.

Now, let's talk about the ZX Spectrum's screen memory. The screen is organized in a peculiar way, and writing to it can be slow, especially when you're dealing with attribute clashes. Attribute clashes occur when you try to draw pixels of different colors within the same 8x8 character block. This forces the Spectrum to do extra work to handle the color changes. To avoid attribute clashes, try to draw your arcs in a way that minimizes color changes within these blocks. Drawing in a single color can often be the fastest approach.

Buffering is another technique that can help with screen-writing performance. Instead of writing pixels directly to the screen memory, we can write them to a buffer in RAM. Once the entire arc is drawn in the buffer, we can then copy the buffer to the screen in one go. This can be more efficient because it reduces the number of individual screen-write operations.

Don't forget about inline assembly. If you're coding in a higher-level language like BASIC or C, you can often embed assembly code directly into your program. This allows you to hand-optimize the most critical parts of your arc-drawing routine, giving you fine-grained control over the Z80's instructions. Inline assembly can be a bit tricky to work with, but it's a powerful tool for squeezing out maximum performance.

And of course, the golden rule of optimization: profile your code. Use a profiler or timing tools to identify the bottlenecks in your routine. Focus your optimization efforts on the parts of the code that are taking the most time. Sometimes, the biggest performance gains come from unexpected places!

Putting It All Together: A Sample Arc-Drawing Routine (Conceptual)

Okay, let's try to piece together a conceptual arc-drawing routine based on everything we've discussed. This won't be a complete, runnable Z80 assembly listing (that would be a whole article in itself!), but it will give you a solid idea of how the pieces fit together.

We'll stick with Bresenham's algorithm because it's a great balance of speed and simplicity. Here's the general flow:

1. Initialization:

   • Set up the circle's center coordinates (x0, y0) and radius (r).
   • Initialize the starting point (x, y) to (0, r).
   • Initialize the error term.

2. Main Loop:

   • Plot the current pixel (x + x0, y + y0) and its seven symmetrical counterparts. (Remember, we're leveraging symmetry!)
   • Check the error term.
   • If the error term is positive, move horizontally (increment x) and adjust the error term.
   • If the error term is negative, move diagonally (increment x and decrement y) and adjust the error term.
   • Repeat until we reach the 45-degree line (x >= y).

3. Plotting Pixels:

   • The core of this routine is the pixel-plotting function. This is where we'll need to consider the ZX Spectrum's screen memory organization and attribute clashes.
   • A basic implementation might directly write the pixel to the screen memory.
   • For better performance, we might use buffering or inline assembly to optimize this step.

4. Symmetry:

   • For each pixel we calculate in the main loop, we need to plot its seven symmetrical counterparts. This involves simple arithmetic to reflect the pixel across the circle's axes and diagonals.
   • This step is crucial for efficiency, as it allows us to draw the entire circle (or arc) by only calculating one octant.

Now, let's add some conceptual Z80 assembly snippets (remember, this is just for illustration!):

; Initialization (very simplified)
LD HL, (circle_center_x) ; Load circle center X coordinate
LD (x0), HL
LD HL, (circle_center_y) ; Load circle center Y coordinate
LD (y0), HL
LD HL, (radius) ; Load radius
LD (r), HL
LD HL, 0 ; x = 0
LD (x), HL
LD HL, (r) ; y = r
LD (y), HL
LD HL, error_initial_value ; Initialize error term
LD (error), HL

main_loop:
; Plot pixels (this is where the magic happens!)
CALL plot_pixels ; A subroutine to handle actual screen writing

; Check error and update x, y, error
; ...

; Check if x >= y (end condition)
; ...

; Jump back to main_loop if not done
; ...

plot_pixels:
; This subroutine would handle the ZX Spectrum's screen memory
; and plot the 8 symmetrical pixels
; ...
RET

This is a super-simplified example, of course. A real-world implementation would involve much more detailed Z80 code, careful handling of memory addresses, and optimization tricks. But hopefully, it gives you a feel for the structure of an arc-drawing routine.

Conclusion: Your Arc-Drawing Adventure Begins!

So there you have it, guys! We've explored the fascinating world of arc-drawing algorithms, from the classic Bresenham's to alternative approaches and optimization strategies. You've got a solid foundation for tackling this challenge on the ZX Spectrum. Drawing arcs efficiently on the Z80 can be tricky, but it's also incredibly rewarding.

Remember, the key is to understand the algorithms, leverage integer arithmetic, and be mindful of the ZX Spectrum's quirks. Don't be afraid to experiment, profile your code, and try different optimization techniques. Every clock cycle you save is a victory!

Now, it's time to roll up your sleeves, fire up your assembler, and start coding! Go forth and create those beautiful curves and circles on your ZX Spectrum. I can't wait to see what you come up with. Happy coding, and I will catch you next time!