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At the present time, the following problems are known in MATLAB Answers itself:
- @doc is presenting messed up text until something is selected
- Symbolic output is not displaying. The work-around is to disp(char(EXPRESSION))
- Near the top of each Question is displayed a link of the most recent activity on the question. The link is normally clickable and takes you directly to the relevant contribution. But at the moment the link does not take you anywhere
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Hello, MATLAB fans!
For years, many of you have expressed interest in getting your hands on some cool MathWorks merchandise. I'm thrilled to announce that the wait is over—the MathWorks Merch Shop is officially open!
In our shop, you'll find a variety of exciting items, including baseball caps, mugs, T-shirts, and YETI bottles.
Visit the shop today and explore all the fantastic merchandise we have to offer. Happy shopping!
Just shared an amazing YouTube video that demonstrates a real-time PID position control system using MATLAB and Arduino.
I was curious to startup your new AI Chat playground.
The first screen that popped up made the statement:
"Please keep in mind that AI sometimes writes code and text that seems accurate, but isnt"
Can someone elaborate on what exactly this means with respect to your AI Chat playground integration with the Matlab tools?
Are there any accuracy metrics for this integration?
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23 Stimmen
Inspired by the suggestion of Mr. Chen Lin (MathWorks), I am writing this post with a humble and friendly intent to share some fascinating insights and knowledge about the Schwarzschild radius. My entry, which is related to this post, is named: 'Into the Abyss - Schwarzschild Radius (a time lapse)'.
The Schwarzschild radius (or gravitational radius) defines the radius of the event horizon of a black hole, which is the boundary beyond which nothing, not even light, can escape the gravitational pull of the black hole. This concept comes from the Schwarzschild solution to Einstein’s field equations in general relativity. Black holes are regions of spacetime where gravitational collapse has caused matter to be concentrated within such a small volume that the escape velocity exceeds the speed of light.
This is a rudimentary scientific post, as the matter of Schwarzschild radius - it's true meaning and function, is a much, much, much-more complex "thing" (not known to us entierly, by the third degre of epistemological explanation(s)).
And, very important is to mention: I am NOT an expert - by any means, on this topic, just a very curious guy, in almost anything, that has to do with science.
Schwarzschild Radius (Gravitational Radius)
The Schwarzschild radius (Rₛ) is the critical radius at which an object of mass 
must be compressed to form a black hole, specifically, a non-rotating, uncharged black hole, known as a Schwarzschild black hole. The Schwarzschild radius 
is given by the formula: 
.
must be compressed to form a black hole, specifically, a non-rotating, uncharged black hole, known as a Schwarzschild black hole. The Schwarzschild radius is given by the formula: . Where: 
.
. Key Characteristics are, that for any mass, if that mass is compressed within a sphere with radius equal to 
, the gravitational field is so strong that not even light can escape, thus forming a black hole. The Schwarzschild radius is proportional to the mass. Larger masses have larger Schwarzschild radii.
, the gravitational field is so strong that not even light can escape, thus forming a black hole. The Schwarzschild radius is proportional to the mass. Larger masses have larger Schwarzschild radii.Example:
For the Sun 
: 
.
: .So, if the Sun were compressed into a sphere with a radius of ~3 km, it would become a black hole!
Stellar-mass Black Holes form from the collapse of massive stars (roughly 
). Their Schwarzschild radius ranges from a few kilometers to tens of kilometers.
). Their Schwarzschild radius ranges from a few kilometers to tens of kilometers. Supermassive Black Holes found at the centers of galaxies, such as Sagittarius A in the Milky Way (
), their Schwarzschild radii span from a few million to billions of kilometers!
), their Schwarzschild radii span from a few million to billions of kilometers! Primordial or Micro Black Holes, are the hypothetical small black holes with masses much smaller than stellar masses, where the Schwarzschild radius could be extremely tiny.
A black hole, in general, is a solution to Einstein’s general theory of relativity where spacetime is curved to such an extent that nothing within a certain region, called the event horizon, can escape.
Types of Black Holes:
1. Schwarzschild (Non-rotating, Uncharged):
- This is the simplest type of black hole, described by the Schwarzschild solution.
- Its key feature is the singularity at the center, where the curvature of spacetime becomes infinite.
- No charge, no angular momentum (spin), and spherical symmetry.
2. Kerr (Rotating):
- Describes rotating black holes.
- Involves an additional parameter called angular momentum.
- Has an event horizon and an inner boundary, known as the ergosphere, where spacetime is dragged around by the black hole's rotation.
3. Reissner–Nordström (Charged, Non-rotating):
- A black hole with electric charge.
- A charged black hole has two event horizons (inner and outer) and a central singularity.
4. Kerr–Newman (Rotating and Charged):
- The most general solution, describing a black hole that has both charge and angular momentum.
Relationship Between Schwarzschild Radius and Black Holes
Formation of Black Holes: When a massive star exhausts its nuclear fuel, gravitational collapse can compress the core beyond the Schwarzschild radius, creating a black hole.
Event Horizon: The Schwarzschild radius marks the event horizon for a non-rotating black hole. This is the boundary beyond which no information or matter can escape the black hole.
Curvature of Spacetime: At distances closer than the Schwarzschild radius, spacetime curvature becomes so extreme that all paths, even those of light, are bent towards the black hole’s singularity.
BTW, the term singularity, scientificaly 😊, means that: we do not have a clue what is really happening right there...
Detailed Properties of Black Holes:
a. Singularity:
At the center of a black hole, within the Schwarzschild radius, lies the singularity, a point (or ring in the case of rotating black holes) where gravitational forces compress matter to infinite density and spacetime curvature becomes infinite. General relativity breaks down at the singularity, and a quantum theory of gravity is required for a complete understanding.
b. Event Horizon:
The event horizon is not a physical surface but a boundary where the escape velocity equals the speed of light. For an outside observer, objects falling into a black hole appear to slow down and fade away near the event horizon due to gravitational time dilation, a prediction of general relativity. From the perspective of the infalling object, however, it crosses the event horizon in finite time without noticing anything special at the moment of crossing.
c. Hawking Radiation: (In the post, I told that there is no radiation - to make it simple, although, there is a relatively newly-found (theoretically) radiation. Truth to be said, some physicists are still chalenging this notion, in some of it's parts...)
Quantum mechanical effects near the event horizon predict that black holes can emit radiation (Hawking radiation), a process through which black holes can lose mass and, over very long timescales, potentially evaporate completely. This process has a temperature inversely proportional to the black hole's mass, making large black holes emit extremely weak radiation. (Very trivialy speaking: the concept supposes that an anti-particle is drawn from the vakum and is anihilated with the black's hole matter (particle), and in the process, the black hole looses mass gradually and proportionally to the released energy - very slowly(!)).
This radiation is significant only for small black holes.
Gravitational Time Dilation (here, as well, things become 'super-weird'...)
Near the Schwarzschild radius, the intense gravitational field leads to time dilation. For an external observer far from the black hole, time appears to slow down for an object moving toward the event horizon. As it approaches the Schwarzschild radius, time dilation becomes so extreme that the object appears frozen in time at the horizon.
The time dilation factor is given by: 
Eg. Approaching the Schwarzschild radius and theoretically remaining just outside of it for a few hours would correspond to the passage of approximately several decades on Earth due to relativistic time dilation.
Using relativistic equations, it's estimated that near the event horizon 2 hours (120 minutes) near the black hole Sagittarius A* (as already mentioned ~ 4 million 
) - in the center of our galaxy Milky Way, could correspond to 83 years passing on Earth! However, this varies based on the precise distance from the event horizon (give or take, a decade 😬).
) - in the center of our galaxy Milky Way, could correspond to 83 years passing on Earth! However, this varies based on the precise distance from the event horizon (give or take, a decade 😬).Information Paradox (definte answer on this question, 'hold's the keys of the universe' 😊, maybe...)
The black hole information paradox arises from the seeming contradiction between general relativity and quantum mechanics.
According to quantum mechanics, information cannot be destroyed, yet anything falling into a black hole seems to be lost beyond the event horizon. Hawking radiation, which allows a black hole to evaporate, does not appear to carry information about the matter that fell into the black hole, leading to ongoing debates and research into how information is preserved in the context of black holes, or not...!
Schwarzschild Radius is the key parameter defining the size of the event horizon of a non-rotating black hole. Black Holes are regions where the Schwarzschild radius constrains all physical phenomena due to extreme gravitational forces, forming event horizons and singularities. The interaction between general relativity and quantum mechanics in the context of black holes (e.g., Hawking radiation and the information paradox) remains one of the most intriguing areas in modern theoretical physics.
For detailed and further reading: https://www.sciencedirect.com/topics/physics-and-astronomy/hawking-radiation.
I hope you will find this post, and information provided, interesting.
Hello! The MathWorks Book Program is thrilled to welcome you to our discussion channel dedicated to books on MATLAB and Simulink. Here, you can:
- Promote Your Books: Are you an author of a book on MATLAB or Simulink? Feel free to share your work with our community. We’re eager to learn about your insights and contributions to the field.
- Request Recommendations: Looking for a book on a specific topic? Whether you're diving into advanced simulations or just starting with MATLAB, our community is here to help you find the perfect read.
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We’re excited to see the discussions and exchanges that will unfold here. Whether you're an expert or beginner, there's a place for you in our community. Let's embark on this journey together!
We are thrilled to announce the grand prize winners of our MATLAB Shorts Mini Hack contest! This year, we invited the MATLAB Graphics and Charting team, the authors of the MATLAB functions used in every entry, to be our judges. After careful consideration, they have selected the top three winners:
Judge comments: Realism & detailed comments; wowed us with Manta Ray
2nd place – Jenny Bosten
Judge comments: Topical hacks : Auroras & Wind turbine; beautiful landscapes & nightscapes
3rd place - Vasilis Bellos
Judge comments: Nice algorithms & extra comments; can’t go wrong with Pumpkins
Judge comments: Impressive spring & cubes!
In addition, after validating the votes, we are pleased to announce the top 10 participants on the leaderboard:
Congratulations to all! Your creativity and skills have inspired many of us to explore and learn new skills, and make this contest a big success!
You can make a lot of interesting objects with matlab primitive shapes (e.g. "cylinder," "sphere," "ellipsoid") by beginning with some of the built-in Matlab primitives and simply applying deformations. The gif above demonstrates how the Manta animation was created using a cylinder as the primitive and successively applying deformations: (https://www.mathworks.com/matlabcentral/communitycontests/contests/8/entries/16252);
Similarly, last year a sphere was deformed to create a face in two of my submissions, for example, the profile in "waking":
You can piece-wise assemble images, but one of the advantages of creating objects with deformations is that you have a parametric representation of the surface. Creating a higher or lower polygon rendering of the surface is as simple as declaring the number of faces in the orignal primitive. For example here is the scene in "snowfall" using sphere with different numbers of input faces:
sphere(100)
sphere(500)
High poly models aren't always better. Low-polygon shapes can sometimes add a little distance from that low point in the uncanny valley.
Go to this page, scroll down to the middle of the long page where you see "Coding Photo editing STEM Business ...." and select "STEM". Voilà!
I was browsing the MathWorks website and decided to check the Cody leaderboard. To my surprise, William has now solved 5,000 problems. At the moment, there are 5,227 problems on Cody, so William has solved over 95%. The next competitor is over 500 problems behind. His score is also clearly the highest, approaching 60,000.
What incredible short movies can be crafted with no more than 2000 characters of MATLAB code? Discover the creativity in our GALLERY from the MATLAB Shorts Mini Hack contest.
Vote on your favorite short movies by Nov.10th. We are giving out MATLAB T-shirts to 10 lucky voters!
Tips: the more you vote, the higher your chance to win.
Mari is helping Dad work.
My favorite image processing book is The Image Processing Handbook by John Russ. It shows a wide variety of examples of algorithms from a wide variety of image sources and techniques. It's light on math so it's easy to read. You can find both hardcover and eBooks on Amazon.com Image Processing Handbook
There is also a Book by Steve Eddins, former leader of the image processing team at Mathworks. Has MATLAB code with it. Digital Image Processing Using MATLAB
You might also want to look at the free online book http://szeliski.org/Book/
Next week is MATLAB EXPO week and it will be the first one that I'm presenting at! I'll be giving two presentations, both of which are related to the intersection of MATLAB and open source software.
- Open Source Software and MATLAB: Principles, Practices, and Python Along with MathWorks' Heather Gorr. We we discuss three different types of open source software with repsect to their relationship to MATLAB
- The CLASSIX Story: Developing the Same Algorithm in MATLAB and Python Simultaneously A collaboration with Prof. Stefan Guettel from University of Manchester. Developing his clustering algorithm, CLASSIX, in both Python and MATLAB simulatenously helped provide insights that made the final code better than if just one language was used.
There are a ton of other great talks too. Come join us! (It's free!) MATLAB EXPO 2024
syms u v
atan2alt(v,u)
function Z = atan2alt(V,U)
% extension of atan2(V,U) into the complex plane
Z = -1i*log((U+1i*V)./sqrt(U.^2+V.^2));
% check for purely real input. if so, zero out the imaginary part.
realInputs = (imag(U) == 0) & (imag(V) == 0);
Z(realInputs) = real(Z(realInputs));
end
As I am editing this post, I see the expected symbolic display in the nice form as have grown to love. However, when I save the post, it does not display. (In fact, it shows up here in the discussions post.) This seems to be a new problem, as I have not seen that failure mode in the past.
You can see the problem in this Answer forum response of mine, where it did fail.
It would be nice to have a function to shade between two curves. This is a common question asked on Answers and there are some File Exchange entries on it but it's such a common thing to want to do I think there should be a built in function for it. I'm thinking of something like
plotsWithShading(x1, y1, 'r-', x2, y2, 'b-', 'ShadingColor', [.7, .5, .3], 'Opacity', 0.5);
So we can specify the coordinates of the two curves, and the shading color to be used, and its opacity, and it would shade the region between the two curves where the x ranges overlap. Other options should also be accepted, like line with, line style, markers or not, etc. Perhaps all those options could be put into a structure as fields, like
plotsWithShading(x1, y1, options1, x2, y2, options2, 'ShadingColor', [.7, .5, .3], 'Opacity', 0.5);
the shading options could also (optionally) be a structure. I know it can be done with a series of other functions like patch or fill, but it's kind of tricky and not obvious as we can see from the number of questions about how to do it.
Does anyone else think this would be a convenient function to add?
At the onset of each week, I release a post that analyzes code with the intent of making it accessible for beginners, while also providing insights that can benefit more experienced users seeking to learn new techniques or approaches.
This week, my inspiration comes from the fractal art produced in MATLAB, as presented in my entry Whispers of the Ocean's Breeze:
Below, I offer a pretty detailed walkthrough of the code break-down, with the goal of creating both an educational and stimulating experience for those eager to learn or find some inspiration. Taking into account that this post is somewhat lengthy, it provides a breakdown and summary of various techniques. It is my hope that it will assist someone and allow readers to focus on the sections that are of most interest to them.
While the code contains comments, this post offers additional explanations and details.
1. Function Definition and Metadata
function drawframe(f)
Line 1: Defines the main function drawframe, which takes a single parameter f. This parameter controls various aspects of the animation, such as movement or speed.
% Audio source: Klapa Šibenik (comp. Arsen Dedić) -
% - Zaludu me svitovala mati
% (Hrvatska 🇭🇷, (Dalmacija))
% Enhanced aesthetics and added dynamic movement,
% offering a creative Remix of my earlier concept.
% (This version brings richer visual appeal,smoother transitions,
% and a more engaging animation flow)
Lines 2-4: Commented-out lines providing metadata or notes about the code. These comments describe the aesthetic goals and improvements in this version, highlighting that it's a remix of one of my earlier entry's, with added dynamic movement and smoother visuals. (These notes are not executed by MATLAB.)
A general tip for using comments: include comments in your code as frequently as needed. They serve as helpful reminders of what each part of the code does, especially when you revisit it after some time, and make it easier for others who may read or use your code.
2. Function Call to seaweed
seaweed(4) % The value in brackets can be adjusted for significantly
% enhanced visualization,
% but it exceeds the 235-second limit in the contest script.
% Feel free to experiment at Desktop workstation - with higher
% values in the loop,
% for more complex and beautiful results.
Line 5: Calls the seaweed function with an argument of 4. The number 4 controls the recursion depth, affecting how detailed or complex the 'seaweed' pattern will be.
Lines 6-9: Comment explaining that increasing the value in seaweed(4) enhances visualization but may exceed time limits in contest environment(s). This suggests adjusting this parameter on a desktop to explore more intricate patterns...
3. Definition of seaweed Function
function seaweed(k)
% Set up the figure window with a specific position and background color
figure('Position', [60+2*f, 60-2*f, 600, 600], 'Color', [0.15, 0.15, 0.5]);
Line 10: Defines the seaweed function, which takes a parameter k (depth of recursion). This function initializes the graphical figure window.
Line 12: Sets up the figure window with a position influenced by f, making the figure’s position change dynamically with f. The background color [0.15, 0.15, 0.5] creates a dark blue background, enhancing the underwater aesthetic.
4. Recursive Drawing with crta Function
crta([0, 0], 90, k, k);
% Make the axes equal and turn them off for a clean figure
axis equal
axis off
Line 14: Calls the recursive function crta, starting the drawing process at [0, 0](origin point) with a 90-degree angle. Both k values set the initial recursion depth and maximum recursion level.
Lines 16-17: Sets the axis scale to equal, ensuring no distortion, and turns off the axes for a cleaner display.
5. Definition of crtaFunction and Initialization of Parameters
function crta(tck, ugao, prstiter, r)
% Define thickness of line segment proportional to current depth
sir = 5 * (prstiter / r);
% Define length of the line segment
duz1 = 5 * prstiter;
Line 18: Defines crta, a nested function within seaweed, taking parameters tck (current coordinates), ugao (angle), prstiter (current recursion depth), and r (maximum recursion depth).
Lines 20-21: Defines sir (line thickness) to be proportional to the current recursion depth, prstiter, creating thinner branches as depth increases. While, duz1 defines the line segment length, which shortens with each recursion, creating perspective.
6. Angle Calculations for Branching
% Define four branching angles with slight variations
ug1 = ugao + 15 + (f / 15);
ug2 = ugao + 7 - f / 15;
ug3 = ugao - 7 + f / 15;
ug4 = ugao - 15 - f / 15;
Lines 23-27: Sets branching angles ug1, ug2, ug3, and ug4 relative to the initial angle ugao, adding and subtracting small amounts. These angles, influenced by f, introduce subtle variations, enhancing the natural appearance.
7. Calculations for Branch Endpoints
% Calculate endpoints of each line segment for the four angles
a1 = duz1 * sind(ug1) + tck(2);
b1 = duz1 * cosd(ug1) + tck(1);
c2 = duz1 * sind(ug2) + tck(2);
a2 = duz1 * cosd(ug2) + tck(1);
b3 = duz1 * sind(ug3) + tck(2);
c3 = duz1 * cosd(ug3) + tck(1);
d4 = duz1 * sind(ug4) + tck(2);
e4 = duz1 * cosd(ug4) + tck(1);
Lines 29-36: Calculates x and y endpoints for each of the four branches using sind and cosd functions, which convert angles into coordinates. Each branch starts at tck(current In this section, the code calculates the endpoints of four line segments, each corresponding to a distinct angle (ug1, ug2,ug3, and ug4). These endpoints are computed based on the length of the line segment duz1, which scales with recursion depth to make each segment shorter as the recursive function progresses. The trigonometric functions sind and cosd are used here to calculate the horizontal and vertical displacements of each segment relative to the current position, tck. While, sind and cosd functions compute the sine and cosine of each angle in degrees, returning the y and x displacements, respectively. For each angle, multiplying by duz1scales these displacements to achieve the intended length for each line segment. Each endpoint coordinate is calculated by adding these displacements to the initial position, tck, to determine the final position for each branch segment:
- a1 and b1 represent the y and x endpoints for the segment at angle ug1
- c2 and a2 represent the y and x endpoints for the segment at angle ug2
- b3 and c3 represent the y and x endpoints for the segment at angle ug3
- d4 and e4 represent the y and x endpoints for the segment at angle ug4
These coordinates form the four main branches radiating out from the current position in different directions. By varying the angle slightly for each branch and scaling the length proportionally, the function generates a visually rich, organic branching structure that resembles seaweed or other natural patterns.
8. Midpoint Calculations for Additional Complexity
% Calculate midpoints for additional "leaves" to
% simulate complexity
uga1 = ug2 - 5 + f / 5;
ugb2 = ug3 + 5 - f / 5;
uga2 = duz1 / 2 * sind(uga1) + c2 + f / 20;
ugb3 = duz1 / 2 * cosd(uga1) + a2 - f / 20;
ugc2 = duz1 / 2 * sind(ugb2) + b3 + f / 20;
ugda1 = duz1 / 2 * cosd(ugb2) + c3 - f / 20;
Lines 38-44: Calculates midpoint angles and positions for extra “leaf” structures. This further enhances the fractal appearance by adding more detail, as these points fall between main branches!
Additional midpoints are calculated to add further detail and complexity to the fractal pattern. These midpoints represent extra branches or “leaves” that emerge from within the main branch segments, enhancing the natural, organic appearance of the fractal structure. Consequently, uga1 and ugb2 are new angles derived by slightly modifying the main branch angles ug2and ug3. The adjustments are made by adding and subtracting small values, including a component based on f. These subtle variations create slight deviations in the angles of the additional branches, making them appear more random and organic, like leaves growing off main stems in varied directions. Once the new angles uga1 and ugb2 are defined, they are used to calculate intermediate coordinates along the main branch lines. These midpoints are positioned halfway along each branch segment, representing the location from which the extra “leaf” branches will emerge.
To find these midpoints:
- uga2 and ugb3 use sind(uga1) and cosd(uga1)to calculate the y and x coordinates halfway along the segment for angle ug2.
- ugc2 and ugda1 similarly use sind(ugb2) and cosd(ugb2) to get the coordinates for angle ug3.
Each midpoint calculation also includes a slight additional offset based on f (like f / 20), adding variation in their positions and contributing to the irregular, natural look of the structure.
By adding these secondary branches, the fractal pattern gains more intricacy. These “leaves” give a more complex and dense appearance, resembling the growth patterns of plants or seaweed where smaller branches diverge from main stems. The addition of midpoints also contributes to the overall depth and richness of the fractal design, ensuring that each recursive call doesn’t simply repeat but also grows in visual detail, making the resulting fractal more visually appealing and realistic. The midpoint calculations thus play a crucial role in enhancing the visual complexity of the fractal by introducing smaller, secondary branches that break up the regularity of the main branches, making the structure more detailed and lifelike.
9. Color Definition Based on Depth
% Define color based on depth, simulating a gradient effect
% as recursion deepens
boja = [1 - (prstiter / r), 1 - 0.5 * (prstiter / r), 0];
Line 46: Defines the color boja as a gradient that shifts from yellow to dark orange based on recursion depth. This gradient effect enhances the visual depth of the pattern.
This code sets up a color gradient for each branch segment based on its recursion depth. This approach not only adds aesthetic appeal but also visually separates different levels of recursion, making it easier to perceive depth within the fractal. The variable boja is an RGB color array, where each element represents the intensity of red, green, and blue respectively, on a scale from 0 (no intensity) to 1(full intensity).
The first element, 1 - (prstiter / r), controls the red component. The second element, 1 - 0.5 * (prstiter / r), controls the green component. The third element is set to 0, meaning there is no blue in the color, resulting in a gradient that shifts from yellow (where both red and green are high) to darker orange and then brownish tones as recursion deepens. The color gradually shifts from a bright yellowish tone at shallow recursion levels to a darker, warmer orange as recursion depth increases. This is achieved by gradually decreasing the red and green components of the color as prstiter (current recursion depth) approaches r (maximum recursion depth). At the top levels of recursion (where prstiter is closer to r), the color becomes darker and more subdued, giving the branches a gradient that makes the structure look natural and complex. This effect is reminiscent of how colors in nature tend to fade or darken with distance or depth, such as in underwater scenes where light penetration decreases with depth.
The gradient serves as a visual cue that helps distinguish between different recursion levels. Since each level is progressively darker, viewers can intuitively sense the depth of each branch, which adds to the three-dimensional effect of the fractal. The use of warm colors (yellow to orange) for each branch segment helps the fractal pattern stand out vividly against the cool blue background set in the seaweed function. This color contrast enhances the underwater, organic look of the structure, making it appear as though the "seaweed" is reaching out toward a light source above. This coloring strategy also contributes to the fractal’s aesthetic complexity. By associating color depth with recursion depth, the fractal appears to have layers, creating a visually satisfying and realistic effect.
10. Plotting Branch Segments
% Plot main branches from the starting point (tck) to the calculated
% endpoints with color and transparency
p1 = plot([tck(1), b1], [tck(2), a1], 'LineWidth', sir, 'Color', boja);
hold on
s2 = plot([tck(1), a2], [tck(2), c2], 'LineWidth', sir, 'Color', boja);
s3 = plot([tck(1), c3], [tck(2), b3], 'LineWidth', sir, 'Color', boja);
s4 = plot([tck(1), e4], [tck(2), d4], 'LineWidth', sir, 'Color', boja);
% Plot secondary branches connecting midpoints for added detail
s5 = plot([a2, ugb3], [c2, uga2], 'LineWidth', sir, 'Color', boja);
s6 = plot([c3, ugda1], [b3, ugc2], 'LineWidth', sir, 'Color', boja);
Lines 48-56: Plots the main branches and secondary branches for added detail. Each plot command connects points with a specified thickness and color, creating the branching effect.
Here presented code, plots the main branches and additional “leaf” segments, giving form to the fractal pattern. Each plot command specifies a line segment by connecting two points, with attributes like line width (sir) and color (boja) enhancing the realism and aesthetic detail. Lines from p1to s4 represent a branch extending outward from the current point tck to its calculated endpoint. The branch segments p1, s2, s3, and s4 form the primary structure of the fractal by branching off at angles ug1, ug2, ug3, and ug4 respectively, calculated in earlier presented and explained steps. The plot command takes a pair of [x, y] coordinates that define the line’s start and end points. For instance, p1 = plot([tck(1), b1], [tck(2), a1], 'LineWidth', sir, 'Color', boja); draws a line from tck(the current position) to (b1, a1), one of the endpoints.
The arguments 'LineWidth', sir and 'Color', boja ensure that each line segment has a thickness and color appropriate to its recursion level, making higher-level branches thicker and more prominent while creating a natural gradient. The command hold on is crucial here, it allows MATLAB to draw multiple line segments within the same figure window without erasing the previous segments. This is necessary for the recursive nature of the fractal, as each call to crtaadds branches to the existing structure, layering them to form a complex, interconnected pattern. Lines s5 and s6 represent additional “leaf”segments, plotted between midpoints calculated in Section 8. These smaller branches diverge from the main branches, adding further intricacy and detail to the fractal. By connecting midpoints (such as a2 to ugb3 and c3 to ugda1), the code generates extra “leafy” offshoots that break up the regularity of the main branches.
These segments make the fractal look more organic, akin to the smaller branches and leaves one might see on real plants or seaweed. Similar to the main branches, the secondary branches use sir and boja for line width and color, ensuring consistent visual depth and blending them seamlessly into the overall pattern. This layering allows the fractal to resemble natural structures like foliage or underwater vegetation. The combination of primary and secondary branches contributes to both symmetry and asymmetry in the fractal. While the primary branches provide a balanced, four-way split, the secondary branches introduce slight irregularities, which lend an organic feel to the pattern. Finally, by plotting each segment separately, the code achieves a highly customizable structure. Line thickness, color, and endpoint coordinates can be easily adjusted for each recursion level, allowing flexibility in the appearance and feel of the fractal.
11. Setting Transparency and Recursion
% Set transparency for each plot segment
s1.Color(4) = 0.95;
s2.Color(4) = 0.95;
s3.Color(4) = 0.95;
s4.Color(4) = 0.95;
s5.Color(4) = 0.95;
s6.Color(4) = 0.95;
% Continue recursive drawing if there are levels left ( prstiter > 0)
if prstiter - 1 > 0
% Recursive calls for each of the main branches with
% updated angles and decreased recursion depth
crta([b1, a1], ug1, prstiter - 1, r);
crta([ugb3, uga2], uga1, prstiter - 1, r);
crta([ugda1, ugc2], ugb2, prstiter - 1, r);
crta([e4, d4], ug4, prstiter - 1, r);
end
end
end
Lines 58-63: Sets the transparency of each branch segment to 0.95, creating a slightly translucent effect.
Lines 65-71: Checks if recursion should continue (i.e., if prstiter > 0). If so, the crta function recursively calls itself with updated angles and positions, generating the next level of branching until prstiter reaches the value of 0.
This code applies transparency to each branch segment to enhance the visual layering effect and initiates further recursion for drawing deeper levels of the fractal. Lines s1.Color(4) = 0.95; through s6.Color(4) = 0.95; apply transparency to each of the plot segments, allowing branches to be slightly see-through. In MATLAB, the fourth element of the Color property, Color(4), represents the alpha (transparency) value. That is, setting it to 0.95 makes each branch segment 95% opaque, meaning it is just translucent enough to create a layered effect where overlapping branches blend slightly. This subtle transparency creates depth, giving the impression that some branches are behind others, which enhances the natural, three-dimensional appearance of the fractal structure. The transparency effect also softens the overall image, making the fractal appear less rigid and more fluid in water.
Line: if prstiter - 1 > 0, checks if further recursion should occur by verifying that prstiter (the current recursion depth) is greater than 1. If prstiter is greater than 1, the function proceeds to recursively call crta, reducing prstiter by 1 with each call. This gradual reduction in prstiter ensures that recursion continues until the maximum depth, defined by r, is reached. As the recursion depth decreases with each call, the branch segments become progressively shorter and thinner, creating a tapered effect that adds to the realistic, fractal-like branching.
Recursive Calls of the function crta, calls itself four times, once for each main branch direction (ug1, ug2, ug3, ug4), using updated coordinates and angles:
- crta([b1, a1], ug1, prstiter - 1, r); initiates a recursive call for the branch at angle ug1.
- crta([ugb3, uga2], uga1, prstiter - 1, r); starts recursion from the midpoint branch at angle uga1.
- crta([ugda1, ugc2], ugb2, prstiter - 1, r); continues recursion from the midpoint branch at angle ugb2.
- crta([e4, d4], ug4, prstiter - 1, r); initiates recursion from the branch at angle ug4.
Each recursive call passes a new starting point (calculated in previous steps) and an adjusted angle. These recursive calls add the next level of branching, gradually building out the entire fractal structure. The recursive calls are fundamental to constructing the fractal pattern. By creating multiple levels of branching, each progressively smaller and more complex, the fractal develops a rich, layered structure that mimics natural growth patterns. The recursive structure also allows for variations in each level, as each branch is influenced by slightly different angles and positions, resulting in an organic, non-uniform look. This natural irregularity is key to creating a visually appealing fractal. Additionally, since each recursive call has transparency applied to its branches, the resulting fractal has a soft, blended appearance. Overlapping branches appear to merge gently, creating a cohesive, three-dimensional visual effect.
End of Code
end
This line closes the entire drawframe function, completing the recursive fractal drawing of the seaweed structure.
Sometimes, neglecting to include the necessary closure for a function can lead to unexpected surprises in the code. Always be vigilant about ensuring that functions, loops, and other structures are properly closed.
Summary: This whole code uses recursion and geometry to create natural-looking, fractal-inspired patterns that mimic the movement and appearance of seaweed, achieving complexity and organic flow through simple recursive structure and dynamic angle variations.
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